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A systematic literature review of Burgers’ equation with recent advances

Authors:
  • Bharata Mata College Thrikkakara
  • Sacred Heart College, Chalakudy, India

Abstract

Even if numerical simulation of the Burgers’ equation is well documented in the literature, a detailed literature survey indicates that gaps still exist for comparative discussion regarding the physical and mathematical significance of the Burgers’ equation. Recently, an increasing interest has been developed within the scientific community, for studying non-linear convective–diffusive partial differential equations partly due to the tremendous improvement in computational capacity. Burgers’ equation whose exact solution is well known, is one of the famous non-linear partial differential equations which is suitable for the analysis of various important areas. A brief historical review of not only the mathematical, but also the physical significance of the solution of Burgers’ equation is presented, emphasising current research strategies, and the challenges that remain regarding the accuracy, stability and convergence of various schemes are discussed. One of the objectives of this paper is to discuss the recent developments in mathematical modelling of Burgers’ equation and thus open doors for improvement. No claim is made that the content of the paper is new. However, it is a sincere effort to outline the physical and mathematical importance of Burgers’ equation in the most simplified ways. We throw some light on the plethora of challenges which need to be overcome in the research areas and give motivation for the next breakthrough to take place in a numerical simulation of ordinary / partial differential equations.
Pramana – J. Phys. (2018) 90:69 © Indian Academy of Sciences
https://doi.org/10.1007/s12043-018-1559-4
A systematic literature review of Burgers’ equation with recent
advances
MAYUR P BONKILE1, ASHISH AWASTHI2,, C LAKSHMI2, VIJITHA MUKUNDAN2and
V S ASWIN2
1Department of Energy Science and Engineering, Indian Institute of Technology Bombay,
Powai, Mumbai 400 076, India
2Department of Mathematics, National Institute of Technology Calicut, Kozhikode 673 601, India
Corresponding author. E-mail: aawasthi@nitc.ac.in
MS received 21 June 2017; revised 30 November 2017; accepted 1 December 2017
Abstract. Even if numerical simulation of the Burgers’ equation is well documented in the literature, a detailed
literature survey indicates that gaps still exist for comparative discussion regarding the physical and mathematical
significance of the Burgers’ equation. Recently, an increasing interest has been developed within the scientific
community, for studying non-linear convective–diffusive partial differential equations partly due to the tremendous
improvement in computational capacity. Burgers’ equation whose exact solution is well known, is one of the
famous non-linear partial differential equations which is suitable for the analysis of various important areas. A
brief historical review of not only the mathematical, but also the physical significance of the solution of Burgers’
equation is presented, emphasising current research strategies, and the challenges that remain regarding the accuracy,
stability and convergence of various schemes are discussed. One of the objectives of this paper is to discuss the recent
developments in mathematical modelling of Burgers’ equation and thus open doors for improvement. No claim is
made that the content of the paper is new. However, it is a sincere effort to outline the physical and mathematical
importance of Burgers’ equation in the most simplified ways. We throw some light on the plethora of challenges
which need to be overcome in the research areas and give motivation for the next breakthrough to take place in a
numerical simulation of ordinary/partial differential equations.
Keywords. Burgers’ equation; non-linear convection–diffusion equation; Hopf–Cole transformation; numerical
solutions.
PACS Nos 12.60.Jv; 12.10.Dm; 98.80.Cq; 11.30.Hv
1. Introduction
The numerical simulation of non-linear convective–
diffusive partial differential equations encountered in
computational fluid dynamics (CFD) has been a sig-
nificant research topic for many decades, both in heat
transfer and in fluid mechanics. CFD has been exten-
sively used for various engineering analysis. Since the
time of Newton, people have been aware of numerical
methods. During this period, various solution meth-
ods of ordinary differential equations (ODEs) and
partial differential equations (PDEs) were conceptu-
ally conceived on paper by the scientific commu-
nity. Owing to extraordinary progress in computational
capacity, computer speed has improved much more
rapidly compared to its cost. A CFD is a powerful
tool for solving a wide variety of modern engineer-
ing problems. CFD provides quantitative as well as
qualitative analysis of the system. It helps in the
simulation of solid–fluid interaction. More and more
complex PDEs are developed to consider various real
life parameters affecting the engineering systems. It
is necessary to have a deep understanding of the
mechanism of fluid flow through a pipe, as it is
extensively used in advanced engineering systems,
industry and our day-to-day life. In human body,
the blood is flowing continuously through the arter-
ies and veins. All the modern biomedical instruments
like artificial hearts and dialysis systems work on
the basis of fluid flow through pipes. In refrigera-
tion and air conditioning application, refrigerants are
flowing through pipes. On a broader scale, PDEs
69 Page 2 of 21 Pramana – J. Phys. (2018) 90:69
Figure 1. Historical milestones in the development of Burgers’ equation.
explaining all these physical processes play important
roles in the design and analysis of modern engineering
systems.
In 1915, Harry Bateman (1882–1946) [1], an English
mathematician, introduced a partial differential equa-
tion in his paper along with its corresponding initial and
boundary conditions given by eqs (1)–(3)
u
t+uu
x=ν2u
x2,0<x<L,0<t, (1)
u(x,0)=ψ(x), 0<x<L,(2)
u(0,t)=ζ1(t), u(L,t)=ζ2(t), 0<t, (3)
where u,x,tand νare the velocity, spatial coordinate,
time and kinematic viscosity, respectively. ψ,ζ1and ζ2
are prescribed functions of variables depending upon
the specific conditions for the problem to be solved.
Later in 1948, Johannes Martinus Burgers (1895–1981)
[24], a Dutch physicist, explained the mathematical
modelling of turbulence with the help of eq. (1). Burgers
went on to become one of the leading figures in the
field of fluid mechanics. To honour the contributions of
Burgers, this equation is well known as the Burgers’
equation. Eberhard Hopf (1902–1983) [5] and Julian
David Cole (1925–1999) [6] independently introduced a
transformation to convert Burgers’ equation into a linear
heat equation and solved exactly for an arbitrary initial
condition. Hence, the transformation is famously known
as the Hopf–Cole transformation given by eq. (4).
u(x,t)=−2νθx
θ,(4)
where θsatisfies the famous heat equation
∂θ
t=ν2θ
x2.(5)
The historical milestones in the development of Burgers’
equation is given in figure 1.
Pramana – J. Phys. (2018) 90:69 Page 3 of 21 69
2. Physical significance
During the last few decades, significant efforts have been
oriented towards the development of robust computa-
tional schemes to handle non-linear PDEs found in fluid
mechanics and heat transfer. One of the most celebrated
equation involving both non-linear propagation effects
and diffusive effects is the Burgers’ equation. Burgers’
equation, being a non-linear PDE, represents various
physical problems arising in engineering, which are
inherently difficult to solve. The simultaneous presence
of non-linear convective term (u(∂u/∂ x)) and diffusive
term (ν(∂ 2u/∂x2)) add an additional feature to the Burg-
ers’ equation. When νapproaches zero, eq. (1) become
inviscid Burgers’ equation, which is a model for non-
linear wave propagation. The unsteady heat equation
without internal heat generation arises in the mathemat-
ical modelling of many physical phenomena occurring
in nature. When uapproaches zero, eq. (1) becomes the
heat equation. When temperature varies with respect to
time in a solid material, there will be a corresponding
variation of the rate of heat transfer within the solid as
well as at its boundary. In many practical engineering
problems, the determination of temperature distribution
is required to calculate the local heat transfer rate, ther-
mal expansion and thermal stress at some crucial loca-
tion. Burgers’ equation, which is a fundamental partial
differential equation from fluid mechanics, proves to be
a very good example because of the following reasons:
1. The exact solution of the partial differential equa-
tion is well known.
2. It can be thought of as a hyperbolic problem with
artificial diffusion for small kinematic viscosity ν
and heat equation for very small u.
3. It can be used in boundary layer calculation for the
flow of viscous fluid.
4. It forms a standard test problem for the PDE
solvers.
5. It is suitable for analysis in various fields like
sedimentation of polydispersive suspensions and
colloids, aspect of turbulence, non-linear wave
propagation, growth of molecular interfaces, lon-
gitudinal elastic waves in isotropic solids, traffic
flow, cosmology, gas dynamics and shock wave
theory.
Hence, a multidisciplinary approach is required to study
Burgers’ equation as shown in figure 2.
2.1 Viscous flow and turbulence
Several important developments have been made in the
field of viscous flow and turbulence describing aero-
dynamic flow at standard temperatures and pressures.
Figure 2. A multidisciplinary approach.
Burgers’ equation, having much in common with the
Navier–Stokes equation, plays a vital role in analysing
fluid turbulence. Murray [7] found that there is an ulti-
mate steady turbulent state. The author concluded that
small disturbances ultimately grow into a single large
domain of relatively smooth flow, accompanied by a
vortex sheet in which strong vorticity is concentrated.
The interface of the control theory and fluid dynamics
has a large number of practical applications includ-
ing feedback control of turbulence for drag reduction.
A non-linear Galerkin’s method was used by Baker
et al [8] to propose a methodology for the synthesis
of non-linear finite-dimensional feedback controllers.
They used one-dimensional Burgers’ equation (6) with
distributed control
U
t=1
Re
2U
z2UU
z+b(z)u(t), (6)
where b(z)is the actuator distribution function. A spa-
tially periodic inviscid random forced Burgers’ equation
in arbitrary dimension and the randomly time-dependent
Lagrangian system related to it was considered by Itur-
riaga and Khanin [9]. Bogaevsky [10] showed that
matter accumulates in the shock discontinuities. Bec
and Kahanin [11] explained Burgers’ turbulence as the
study of solutions to the one- or multidimensional Burg-
ers’ equation with random initial conditions or random
forcing. The study of random Lagrangian systems, of
stochastic partial differential equations and their invari-
ant measures, the theory of dynamical systems, the
applications of field theory to the understanding of dissi-
pative anomalies and of multiscaling in hydrodynamic
turbulence have benefited significantly from progress
in Burgers’ turbulence. Recently, it become a point of
attraction to many researchers due to the new emerging
application of a Burgers’ model to statistical physics,
cosmology and fluid dynamics. Navier–Stokes turbu-
lence has one important property: sensitivity to small
perturbations in the initial data and thus the spontaneous
rise of randomness by chaotic dynamics.
The results obtained by Chekhlov and Yakhot [12]
in a simple one-dimensional system, were similar
69 Page 4 of 21 Pramana – J. Phys. (2018) 90:69
to the outcome of experimental investigations of the
real-life three-dimensional turbulence. They considered
the dynamics of velocity fluctuations driven by a white-
in-time random force, in the one-dimensional Burgers’
equation, which showed a biscaling behaviour [13].
WKB calculations are very useful for understanding the
behaviour of the randomly driven Burgers’ equation.
The instanton solution, which describes the exponential
tail of the probability distribution function of veloc-
ity differences for large positive velocity differences,
was found by Gurarie and Migdal [14]. The properties
of the probability density function of velocity differ-
ences were investigated for the three different cases by
Yakhot and Chekhlov [15]. In turbulence, the deviation
of the probability density function tails from Gaussian
was regarded as a manifestation of intermittency [16].
Particular interest, of solutions, of the randomly forced
Burgers’ equation, were the asymptotic properties of
probability distribution functions associated with veloc-
ity gradients and velocity increments. Weinan et al [17]
studied the asymptotic behaviour of the probability dis-
tribution of velocity gradients and velocity increments
using a new and direct approach for analysing the scal-
ing properties of the various distribution functions for
the randomly forced Burgers’ equation.
The randomly forced Burgers’ equation, which is
periodic in xwith period 1, and with white noise
in t, is a prototype for a very wide range of prob-
lems in non-equilibrium statistical physics, where strong
non-linear effects are present [18]. Bec et al [19]pre-
sented the fast Legendre transform numerical scheme
for space-periodic kicked Burgers’ turbulence with
spatially smooth forcing. Bec et al [20] had shown theo-
retically and numerically that the shocks have a universal
global structure which was determined by the topology
of the configuration space. They studied the dynam-
ics of the multidimensional randomly forced Burgers’
equation in the limit of vanishing viscosity. Bec and
Khanin [21] studied the inviscid randomly forced Burg-
ers’ equation with non-periodic forcing on the whole
real line started at t=−. They introduced varia-
tional approach to Burgers’ turbulence and presented
results on the existence and uniqueness of the main
shock and the global minimiser. Gomes et al [22]pre-
sented a very simple and soft approach, which allows to
construct stationary distributions for randomly forced
viscous Hamilton–Jacobi and Burgers’ equations and
proved convergence of this distributions in the limit of
vanishing viscosity. The global solutions satisfy uniform
Lipschitz and semiconvexity properties. The nature of
the fluctuational mechanism is very general, but it guar-
antees only a very slow rate of convergence. It was used
for the uniqueness of global solutions and convergence
of the solutions to the Cauchy problems.
2.2 Shock theory
Burgers’ equation is similar to the Navier–Stokes
equation without the pressure term. For low kine-
matic viscosity, there can be velocity discontinuities, i.e.
shocks. From a physical point of view, viscous model
better explains what happens in actual than in inviscid
equation. Kreiss and Kreiss [23] considered the viscous
Burgers’ equation with the initial and non-homogeneous
Dirichlet boundary conditions to study the effect of
shock on the convergence of steady-state solution.
ut+1
2(u2)x=uxx +f(x),
t0,0x1, > 0,(7)
u(x,0)=g(x), (8)
u(0,t)=a,u(1,t)=b(9)
and the corresponding steady-state problem
1
2(y2)x=yxx +f(x), 0x1, > 0,(10)
y(0)=a,(11)
y(1)=b.(12)
They investigated uniqueness, the existence and prop-
erties of the steady-state solution. It was proved that the
steady-state problem has a unique solution and for suffi-
ciently large , the steady-state equation has a solution.
Moreover, steady-state equation has a unique solution
for all >0. To speed up the convergence to steady
state, the shock should be located at the boundary. On
the other hand, location of shock in the interior leads to
very slow convergence. The speed of convergence was
studied by analysing the corresponding eigenvalue prob-
lem. The non-linear Burgers’ equation was solved by
using shock-capturing schemes for problems involving
formation and propagation of shocks, shock collisions
and expansion of discontinuities [24].
Reyna and Ward [25] were motivated by the work of
Kreiss and Kreiss on the initial boundary value problem
for Burgers’ equation. They investigated the shock layer
behaviour associated with the following viscous shock
problem in the limit 0:
ut+[f(u)]x=uxx,0<x<1,t>0,(13)
u(x,0)=u0(x), u(0,t)=α, u(1,t)=−α. (14)
Here αis a positive constant and the convex non-
linearity f(u)satisfies f(0)=f(0)=0,uf(0)>0
for u= 0and f) =f(α). For Burgers’ equation
f(u)=u2/2. Their main goal was to analyse the slow
shock layer motion for eq. (13) analytically in the limit
0. Analytical explanation for the exponentially
slow phase where the shock layer drifts to its equilibrium
point was given. The method of matched asymptotic
Pramana – J. Phys. (2018) 90:69 Page 5 of 21 69
expansions (MMAE) was used to construct equilibrium
solution to Burgers’ equation. From symmetry, the equi-
librium shock layer solution for eq. (13)isatx=1/2.
Using MMAE, they obtained one-parameter family of
approximate solutions parametrised by x0.
u=−αtanh[α1(xx0)/2]x0.(15)
In the case of equilibrium problem, to eliminate the inde-
terminacy in the MMAE solution for the location of the
shock layer, projection method was used. Burgers’ equa-
tion is applicable for time-dependent problems. Hence,
they used projection method to derive an equation of
motion for the trajectory of the slow moving shock layer.
The result shows that the solution to eq. (13)isgiven
asymptotically by u∼−αtanh[α1(xx0(t))/2], dur-
ing the slow evolutionary phase,where x0(t)satisfies the
ODE
dx0
dt=α[eα1x0eα1(1x0)].(16)
Their main contribution is to show that even if the bound-
ary operator is changed by small value it may destabilise
the equilibrium solution. It results in hitting the shock
layer to one of the physical boundaries instead of drifting
to its equilibrium position. Numerical method based on
preliminary WKB-type transformation of eq. (13)was
formulated to study the shock layer behaviour for 1.
It is advantageous because it magnifies the exponentially
weak interactions and exponentially long time-scale
associated with the shock layer motion becomes trans-
parent. Simple finite-difference method can be used on
the well condition equilibrium transformed problem to
study the shock layer behaviour numerically for rather
small .
2.3 Gas dynamics
The Burgers’ equation shows the interplay of convection
and diffusion present in viscous fluid flow engineering
problem [26]. Also, Burgers’ equation with source terms
appeared in the aerodynamics theory [27].
ut+uux=νuxxλu.(17)
The contribution was the manipulation of a mathemat-
ical problem from the gas dynamics theory, which was
related to the heat exchange in the boundary layer. Heat
can be released in physical application of gas dynamics,
which can be presented in chemically reacting bound-
ary layer or in the propagation of the detonation wave.
Near-equilibrium boundary layers of a gas by present-
ing heat source terms can be considered using Burgers’
equation, i.e.,
ut+uux=νuxxλu;x0,t00,(18)
u(0,x)=φ(x), u(t,0)=f(t). (19)
The term λuin eq. (18) represents the heat released
in the boundary layer. Analytical solution in paramet-
ric form of the Burgers’ equation with source term was
constructed. Lagrange subsidiary equations and split-
ting of the second-order non-linear PDE were used for
constructing solutions. As a result, Monge’s equation
was generated.
2.4 Cosmology
The expanding Universe is irregular and clumpy com-
pared to its uniform initial state [28]. Three-dimensional
Burgers’ equation describes the characteristics of a ran-
dom potential vector field, which was investigated by
Gurbatov and Saichev [29]. In this paper, they dis-
cussed the possible relationship between the large-scale
structure of the Universe and the cellular structure. The
three-dimensional Burgers’ equation
Vt+(V)V=νV(20)
V(r,t=0)=V0(r)(21)
may also be an acceptable model for the turbulence
theory. The way the field V(r,t)behaves described by
eq. (20) is similar to the growth of the Universe lies in the
fact that the field V(r,t)acquires in the time a universal
cellular structure, the presence and reason of formation.
As per the author, one of the mechanisms of formation
of cellular structure of the Universe, the inertial insta-
bility, can be described by analysing the dynamics and
statistics of the fields described by eq. (20). They inves-
tigated the dynamics and statistics of random potential
fields, described by eq. (20). The possible connection of
their characteristics with the large-scale structure of the
Universe was discussed. The adhesion model of large-
scale distribution of matter in the Universe is formulated
by Burgers’ turbulence [3032].
Peebles [33] explained that right after the decoupling
between baryons and photons, the promotive Universe
is a rarefied medium without pressure composed mainly
of non-collisional dust interacting through Newtonian
gravity. The initial density of this dark matter fluctua-
tion is responsible for the formulation of the large-scale
structure in which both the dark non-baryonic matter and
the luminous baryonic matter concentrate. A hydrody-
namical formulation of the cosmological problem leads
to a description where matter evolves with a velocity ¯
V.
Molchanov et al [34] studied the dynamics of the struc-
ture of shock fronts in the inviscid non-homogeneous
Burgers’ equation in Rdin the presence of random forc-
ing due to a degenerate potential. The influence of the
clusters through their gravitational field was considered
69 Page 6 of 21 Pramana – J. Phys. (2018) 90:69
by Burgers’ equation with external force field
Fas given
by eq. (22). The development of cluster is a relative slow
process.
v
t+(v)v=1
2μ2v
F,vR3.(22)
2.5 Traffic flow
Burgers’ equation shows a strong connection between
fluid dynamics and traffic flow models. In the theory
of traffic flow, use of Greenshields [35] model leads to
the formation of Burgers’ equation. Traffic current on
expressway is defined as the number of cars passing at
a reference point per unit time [36]. If vehicle concen-
tration is linearly related to the drift speed in the one-
dimensional case, then vehicle concentration on the road
can be explained by Burgers’ equation in a moving frame
of reference [37]. From the sophisticated level of fluid-
dynamical approach, no traffic jam forms spontaneously
from a state of uniform density. Leibig [38] has studied
how a random initial distribution of steps in the density
profile evolves with time. If we treat traffic as an effec-
tively one-dimensional compressible fluid, then in anal-
ogy with the hydrodynamic theory of fluids, a ‘macro-
scopic’ theory of traffic can be developed [39]. Traffic
flow can be treated as an effectively one-dimensional
compressible fluid (a continuum) when viewed from a
long distance [40]. Second-order model of very light
traffic flow is given by Aw and Rascle [41]. The den-
sity wave in traffic flow, which gradually changes from
non-uniform to uniform distribution, is described by
Burgers’ equation [42]. Equation (23) is considered
as one-dimensional non-linear non-homogeneous Burg-
ers’ equation, which is applicable to other physical
phenomena such as design of feedback control [43],
electrohydrodynamic field in plasma physics [44]and
wind forcing the build-up of water waves [45].
ut+uux=νuxx +F(x,t). (23)
2.6 Quantum field
The Burgers’ equation is the simplest example where
a quantum computer was demonstrably more efficient
at numerically predicting the time-dependent solutions
[46]. The Burgers’ equation is derived as an effective
field theory governing the behaviour of the quantum
computer at its macroscopic scale, where both the lattice
cell size and the time step interval become infinitesimal.
A microscopic scale algorithm for a type-II quantum
computer was presented for modelling the time evolu-
tion of a continuous field governed by the non-linear
Burgers’ equation in one spatial dimension in [47].
The quantum model is a system of qubits, where there
exists a minimum time interval in the time-dependent
dynamics. The measurement steps are dispersed periodi-
cally in time and across all the elements of the quantum
system. Yepez [48] presented an analysis of an open
quantum model of the time-dependent evolution of a
flow field governed by the non-linear Burgers’ equation
in one spatial dimension. This allows to use the quan-
tum algorithm for modelling highly non-linear shock
formation, even severely under-resolved shock fronts,
without the model breaking down. The flow field evolv-
ing under the Burgers’ equation develops sharp features
over time. Therefore, it is a better test of liquid-state
nuclear magnetic resonance (NMR) implementations of
type-II quantum computers than examples using the dif-
fusion equation. The practicality of implementing the
quantum lattice gas (QLG) algorithm using a spatial
NMR technique was proved by the numerical data and
the exact analytical results for the non-linear Burgers’
equation [49].
3. Further significance
The Burger’s equation has been used as a benchmark
problem in parallel and distributed numerical computa-
tion, i.e., algorithms can be tested with the help of known
analytical solutions of Burger’s equation. An OpenMPI-
based hybrid space–time parallel algorithm is discussed
in [50]. GPU-based parallel computing algorithm for the
numerical solution of one-dimensional Burger’s equa-
tion is presented in [51]. In [52], CUDA Fortran was
used for the problem presented in [51]. A parallel com-
puting technique for two-dimensional Burger’s equation
is discussed in [53] with OpenMPI and GPU implemen-
tation. To find solutions of non-linear partial differential
equations, Puffer et al [54] have used the cellular neural
network and Hayati and Karami [55] have used feedfor-
ward neural network learning algorithms. It was tested
and validated for Burgers’ equation.
Zueco [56] has introduced a new idea for the simu-
lation of Burger’s equation through network simulation
method based on an electrical motion analogy. It was
described that the discretised equation was analogous
to an electric network model, and hence, solved using
an electrical circuit simulation software called Pspice. In
a study by Hetmanczyk and Ochs [57], inviscid as well
as the viscous Burger’s equations were simulated using
wave digital simulation technique. First, they derived
an electrical circuit corresponding to the equations and
then found wave-digital realisation by using a circuit
equivalent to a Gears 2-step backward differentiation
formula. The results were compared with exact solution
and solutions from the trapezoidal rule. They found that
Gears method handles discontinuities efficiently.
Pramana – J. Phys. (2018) 90:69 Page 7 of 21 69
Figure 3. Process map for numerical simulation.
In 1995, Esipov [58] derived and studied the physics
of coupled Burgers’ equation and considered it as a
simple model of sedimentation or evolution of scaled
volume concentrations of two kinds of particles in fluid
suspensions or colloids under the effect of gravity.
The study of the fluctuations of concentration during
sedimentation may lead to an experimental realisa-
tion of Burgers’ turbulence. Tsai et al [59]employed
inviscid Burgers’ equation for the validation of a paral-
lel domain-decomposed Chebyshev collection method
developed for atmospheric model simulations. In [60],
Kalman filter of Burgers’ equation is used to examine
the problems associated with atmospheric data assim-
ilation and numerical weather prediction. The forced
Burgers’ equation is considered in [61] for studying vari-
ous errors encountered in the simulations of atmospheric
flow. Propagation of high-intensity noise in the atmo-
sphere can be simulated using Burgers’ equation [62],
and this is the key idea used for the prediction of propa-
gation of noise from jet aircraft and helicopters [6365].
A recent study [66] describes Burgers’ equation as a
mathematical model for one-dimensional groundwater
recharge by spreading.
Control problems have broad applications in the real
life, but introducing a control variable in the governing
equation such as non-linear partial differential equa-
tion like Burgers’ is not easy. Plenty of articles are
available for the control problems in Burgers’ equa-
tion, boundary control techniques, and its numerical
methods are discussed in [6769] and various dis-
tributed control methods can be found in [7072].
Smaoui [73] studied boundary and distributed control
of Burgers’ equation and conducted numerical and ana-
lytical stability analysis. Glass and Guerrero [74]have
proposed a boundary control for the viscous Burgers’
equation for small kinematic viscosity, and proved the
exact controllability property to a non-zero steady-state
situation.
4. Mathematical significance
Consequently, numerical solution of the PDEs has been
a significant research topic for many decades, in both
heat transfer and fluid mechanics. Process map for
numerical simulation is shown in figure 3. First step
in this process is to understand the physics of the
problem, which lead to the formation of a mathemat-
ical model with the help of equations. In most of the
cases, these equations are either ODEs or PDEs. Some
assumptions have to be made, because the real life prob-
lems in engineering are a bit complex to analyse. After
this, these equations are solved by numerical methods
like finite-difference method, finite-element method and
finite-volume method.
Rodin [75] studied some approximate and exact solu-
tions of boundary value problem for Burgers’ equation
with the help of Hopf–Cole transformation. Benton
and Platzman [76] have given 35 distinct analytical
solutions of Burgers’ equation with different initial con-
ditions. Wolf et al [77] discovered a procedure to extend
the analytical solution of Burgers’ equation to the n-
dimensional problems by employing group actions on
coset bundles. Nerney et al [78] extended the solutions
to the curvilinear coordinate systems. Kudryavtsev and
Sapozhnikov [79] proposed a method to find exact solu-
tion of inhomogeneous Burgers’ equation using Hopf–
Cole and Darboux transformations. In [80], analytical
solutions of (1+n)-dimensional Burgers’ equation have
been computed using various semianalytical methods
like homotopy perturbation method, Adomian decom-
position method and differential transform method. The
closed form of the solution has been successfully com-
puted for (1+n), (1+3)and (1+2)dimensions with
an initial condition equal to the sum of its spatial coordi-
nates, i.e., u(x1,x2,...,xn,t=0)=x1+x2+···+xn
and for (1+1)dimension with an initial condition
u(x,t=0)=2x.In[81], Elzaki transformation and
69 Page 8 of 21 Pramana – J. Phys. (2018) 90:69
homotopy perturbation method have been combined to
get a semianalytic solution of Burgers’ equation. Vari-
ous numerical schemes and a few of the commonly used
analytical solutions are listed here.
4.1 Solution for a smooth initial condition
u(x,t)=2πν
L
n=1Cnexp(n2π2νt/L2)nsin(nπx/L)
C0+
n=1Cnexp(n2π2νt/L2)cos(nπx/L).
(24)
Cole [6] has introduced this analytical solution (eq. (24))
of Burgers’ equation (eq. (1)) for a periodic initial distur-
bance ψ(x)and boundary conditions ζ1(t)=ζ2(t)=0.
C0and Cnare the Fourier coefficients and Cole [6]has
also provided a formula to compute coefficients for an
arbitrary periodic initial condition. For a sinusoidal ini-
tial condition (25), these coefficients can be determined
using eqs (26)and(27).
ψ(x)=u0sinπx
L,(25)
C0=1
LL
0
expu0L
2πν 1cos πx
Ldx,(26)
Cn=2
LL
0
expu0L
2πν 1cos πx
L
×cos nπx
Ldx.(27)
The solution corresponding to this initial condition is
plotted in figure 4Aforu0=1andν=0.01 over a
space domain 0 x1.
ψ(x)=u0x(Lx).(28)
For an initial condition like eq. (28), the coefficients C0
and Cncan be determined as
C0=1
LL
0
expu0x2(3L2x)
12νdx,(29)
Cn=2
LL
0
expu0x2(3L2x)
12ν
×cos nπx
Ldx.(30)
Physical behaviour of the solution corresponding to this
initial condition is plotted in figure 4Bforu0=4and
ν=0.1 over a space domain 0 x1.
Figure 4. Physical behaviour of Burgers’ equation for
smooth initial conditions.
Wood [82] has introduced a closed form of exact
solution for the Burgers’ equation over the domain
0x1andt0, in the form
u(x,t)=2πν sinx)exp(π2νt)
α+cosx)exp(π2νt).(31)
Initial and boundary conditions can be derived from the
exact solution (eq. (31)). Figure 4C shows the surface
plot of u(x,t)for α=2andν=0.05. Cecchi et al
[83] have considered an analytical solution analogous to
the sinusoidal wave propagation in a viscous medium as
given in eq. (32) for a homogeneous boundary condition
and periodic initial condition ψ(x)=u(x,t=0)over
0x2.
u(x,t)=2πν sinx)exp(π2ν2t)+4sin(2πx)exp(4π2ν2t)
4+cosx)exp(π2ν2t)+2cos(2πx)exp(4π2ν2t).(32)
For μ=0.01, the behaviour of u(x,t)is plotted in
figure 4D.
4.2 Shock and travelling wave solutions
u(x,t)=Uu1tanh u1(xx1Ut)
2ν.(33)
Equation (33) represents a shock wave with a velocityU.
u1and x1are constants, such that u1=u(−∞,0),u2=
u(+∞,0)and u1>u2,x1=xat which discontinuity
occurs. This problem is presented in Cole’s study [6]and
he mentioned that Bateman [1] has used eq. (33) with
U=0 as a steady-state solution. Many researchers have
been using a simplified version of eq. (33)bymaking
U=u1=λ/2andx1=0 as given in eq. (34). Equation
(34) is plotted in figure 5Aforλ=1andν=0.01.
Pramana – J. Phys. (2018) 90:69 Page 9 of 21 69
Figure 5. Examples of travelling wave and shock wave solu-
tions of viscous Burgers’ equation.
u(x,t)=λ
21+tanhλ
8νt2x).(34)
Christie et al [84] constructed a travelling wave
analytical solution for Burgers’ equation in the form
u(x,t)=μ+α+α) exp(η)
1+exp(η),(35)
η=α(xμtβ)
ν,(36)
where α,βand μare constants, μis the speed of the
wave and βrepresents the point in xat which initial
discontinuity occurs. In most of the articles, values of
these constants were chosen as α=0.4, β=0.125
and μ=0.6. The behaviour of the solution (eq. (35))
corresponding to these constants and ν=0.01 is given
in figure 5B. An example of shock wave (eq. (37)) can
be found in [85] and its surface plot is given in figure 5C
for ν=0.005.
u(x,t)=x/t
1+(t/t0)1/2exp(x2/4νt),t1,(37)
t0=exp1
8ν.(38)
Another interesting shock wave solution of Burgers’
equation is given in eq. (39) and in figure 5Dforν=0.5.
u(x,t)=0.1eA+0.5eB+eC
eA+eB+eC,(39)
A=0.05
ν(x0.5+4.95t),(40)
B=0.25
ν(x0.5+0.75t),(41)
C=0.5
ν(x0.375).(42)
Figure 6. Examples of shock wave and rarefaction wave
solutions of inviscid Burgers’ equation.
The initial and boundary conditions related to the exact
solutions considered in this section can be deduced from
the exact solution itself, by proper substitution of xand
tvalues. For more travelling wave solutions, refer to
Salas [86].
4.3 Shock and rarefaction waves in inviscid Burgers’
equation
Consider Riemann-type initial conditions as given in
eq. (43), such that u1= u2at x=0. Inviscid Burgers’
equation possesses a shock or rarefaction wave solution
depending on whether u1>u2or u1<u2in eq. (43).
For u1=1andu2=0, the solution is given in eq. (44)
and S=1/2 represents the shock speed. The shock
propagation is captured in figure 6A.
ψ(x)=u1,x0,
u2,x>0,(43)
u(x,t)=1,xSt <0,
0,xSt >0.(44)
When u1=0andu2=1, inviscid Burgers’ equation
has a rarefaction wave solution given in eq. (45)and
its propagation is captured in figure 6B. Simulation of
the inviscid Burgers’ equation is a challenge for many
numerical schemes because of the presence of disconti-
nuity in the solution.
u(x,t)=
0,(x/t)<0
x/t,0<(x/t)<1.
1,(x/t)>1
(45)
4.4 Finite-difference method (FDM)
Various methods proposed for mathematical modelling
have their own advantages and disadvantages. Out of
these, FDM, the discretisation method, is the most
simple and oldest method to solve ODEs/PDEs. In
1983, Fletcher [87] gave exact solutions of some
specified two-dimensional Burgers’ equation based on
69 Page 10 of 21 Pramana – J. Phys. (2018) 90:69
two-dimensional Hopf–Cole transformation. Unlike the
one-dimensional Burgers’ equation, two-dimensional
Hopf–Cole transformation cannot be used to convert
two-dimensional Burgers’ equation into a linear heat
equation. For using two-dimensional Hopf–Cole trans-
formation, the condition of potential symmetry must
be satisfied by the two-dimensional Burgers’ equation.
The characteristics of Burgers’ equation was studied
by Aref and Daripa [88] using phase plane analysis.
They have applied the finite-difference method for the
semidiscretisation of space variable over a few grid
points resulting in a system of coupled ODEs. Phase
plane analysis of these systems of ODEs has been
conducted to study the characteristics of Burgers’ equa-
tion. Kutluay et al [89] used Hopf–Cole transformation
to convert Burgers’ equation to heat equation. The
transformed heat equation with the insulated bound-
ary conditions was solved by explicit and exact-explicit
finite-difference method. In 2003, Bahadir [90]pro-
posed a fully implicit finite-difference scheme, while
the non-linear system is solved by Newton’s method.
Hassanien et al [91] developed a two-level three-point
finite-difference scheme, which was fourth-order accu-
rate in space and second-order accurate in the time.
Stability using von-Neumann stability analysis showed
that the method is unconditionally stable. Kadalbajoo
et al [92] developed an implicit scheme for solving
the Burgers’ equation. They used a standard backward
Euler scheme with constant time step to discretise in
the temporal direction and a standard upwind finite-
difference scheme to discretise in spatial direction on
piecewise uniform mesh. The quasilinearisation pro-
cess was used to tackle non-linearity. The sequence of
solutions of the linear equations obtained after applying
quasilinearisation was shown to converge quadratically
to the solution of the original non-linear problem. A
numerical method based on Crank–Nicolson scheme
was put forward by Kadalbajoo and Awasthi [93],
where they first reduced the Burgers’ equation to a
linear heat equation using Hopf–Cole transformation
and then decsretised using Crank–Nicolson scheme.
The mesh size could be chosen without any restric-
tion. It was shown that the scheme was second-order
accurate in both space and time and also uncondi-
tionally stable. Liao [94] proposed a method which
transforms the original non-linear Burgers’ equation
into a linear heat equation using Hopf–Cole transfor-
mation, and transforms the Dirichlet boundary con-
dition into the Robin boundary condition. The linear
heat equation is then solved by an implicit fourth-
order compact finite-difference scheme. Descretisation
in temporal direction was performed using Crank–
Nicolson scheme and the accuracy was improved by
Richardson extrapolation. Comparison proved the supe-
riority of the method over the existing schemes. A
second-order accurate difference scheme is discussed
[95] in which the non-linear system is solved by both
Newton’s method and predictor–corrector method. The
authors have also presented the uniqueness of the dif-
ference solution, the stability and L2convergence of
the difference scheme by the energy method. Pandey
et al [96] tried another method by reducing Burgers’
equation to the heat equation and applying Douglas
finite-difference scheme on the reduced equation. The
method was shown to be unconditionally stable, fourth-
order accurate in space and second-order accurate in
time. The modified local Crank–Nicolson (MLCN)
method for one- and two-dimensional Burgers’ equa-
tions was presented in [97]. The MLCN is an explicit
difference scheme with simple computation and is
unconditionally stable. Liao [98] extended the method in
[94] to solve two-dimensional Burgers’ equation using
an unconditionally stable, compact fourth-order finite-
difference scheme. The author has assumed potential
symmetry condition and hence used two-dimensional
Hopf–Cole transformation to convert non-linear Burg-
ers’ equation into two-dimensional linear heat equa-
tion. The linear heat equation was then solved by an
implicit fourth-order compact finite-difference scheme.
The author has also developed a compact fourth-order
formula to approximate the boundary conditions of
the heat equation, while the initial condition for the
heat equation was approximated using Simpson’s rule.
A semi-implicit finite-difference method was used to
find the numerical solution of two-dimensional coupled
Burgers’ equation in [99]. In 2012, Mousa et al [100]
proposed combined compact finite-difference scheme
for the treatment of one-dimensional Burgers’ equation.
They have used Hopf–Cole transformation to convert
non-linear Burgers’ equation to heat equation. They
have implemented a compact finite-difference scheme
to approximate the space derivatives and a low storage
Runge–Kutta scheme to approximate the time integra-
tion. In [101], Kweyu et al have generated three sets
of varied initial and boundary conditions from gen-
eral analytic solution obtained by using Hopf–Cole
transformation and method of separation of variables.
Numerical solution was obtained by using Crank–
Nicolson scheme and explicit scheme. The accuracy
in terms of consistency, convergence and stability was
determined by means of L1error. Another scheme using
Crank–Nicolson method was discussed by Wani and
Thakar [102], where they approximated utby forward
difference and uuxby central difference at t=tnand
t=tn+1. The scheme remained linear in uat t=tn+1
and νuxx was approximated by usual Crank–Nicolson
expression. The results came out to be better than
those by Kadalbajoo and Awasthi [93]. Srivastava et al
[103] proposed an implicit logarithmic finite-difference
method, for the numerical solution of two-dimensional
Pramana – J. Phys. (2018) 90:69 Page 11 of 21 69
time-dependent coupled viscous Burgers’ equation on
the uniform grid points. Also in [104], an implicit
exponential finite-difference scheme has been proposed
for solving two-dimensional non-linear coupled viscous
Burgers’ equations with appropriate initial and bound-
ary conditions.
4.5 Method of lines (MOL)
In 1930, Rothe [105], who was from former Soviet
Union, introduced method of lines (MOL) in his paper.
MOL does not involve the discretisation of all variables.
MOL is used
to convert the system of PDE into ODE initial value
problem,
to discretise the spatial derivatives together with the
boundary conditions,
to integrate the resulting ODEs using a sophisticated
ODE solver, which takes the burden of time discreti-
sation by choosing the time steps to maintain the
accuracy and stability of the evolving solution.
The MOL semidiscretisation approach was used to
transfer
ut+uux=νuxx (46)
into a system of first-order linear ODE. In the MOL
approach, the ODEs are integrated directly with a stan-
dard code for the task. The stability analysis of the MOL
is the most important and critical factor in their solu-
tion. Stability analysis includes the study of numerical
solution as it gives the means by which the step size
and the numerical integration scheme could be selected.
The stability analysis of MOL for discretisation that
produce initial value type in ODE is easy compared
to the analysis of MOL for discretisation that produce
boundary-valued problems.
The relative merits and demerits of MOL with an
ordinary differential equation solver to classical explicit
and implicit finite-difference techniques were compared
by Kurtz et al [106]. Shampine [107] investigated
factors influencing the choice of ODE solver for the
numerical solution of PDEs by MOL. The numeri-
cal solution of the advection–diffusion equation was
studied. The system arising from the solution of the
advection–diffusion equation by MOL has a Jacobian
with a very simple form. A novel approach to the
development of the code was given by Oymak and Sel-
cuk [108], which involves coupling between MOL and
a parabolic algorithm. This code removes the neces-
sity of iterative solution on pressure and solution of a
Poisson-type equation for the pressure. The main con-
tribution of this paper is the proposal of a time-accurate
Navier–Stokes code based on the MOL approach with
a non-iterative algorithm for the pressure. The spatial
derivatives of dependent variables were approximated
by using a 5-point Lagrange interpolation polynomial.
Convective terms were discretised using upwinds and
diffusion terms were discretised centrally. 2D Navier–
Stokes equation was used by Selcuk et al [109] to test the
performance of MOL and FDM for checking solution
accuracy and CPU time. The results were compared with
the previously reported results in literature. A parabolic
algorithm was used, which removed the necessity of
iterative solution and did not require the solution of a
Poisson-type equation for the pressure. The method was
used to calculate
axial velocity and pressure distribution in pipe flow,
steady-state reattachment lengths in sudden expan-
sion flow on uniform grid distribution.
Javidi [110] presented a new method for solving the
Burger’s equation by combination of method of lines
(MOL) and matrix-free modified extended backward
differential formula (MF-MEBDF), where a difference
scheme of O(h4)was used to approximate u(x,t)
and uxx(x,t). The resulting set of ordinary differen-
tial equation in twas solved by MF-MEBDF, where
an exact Newton method and then the IOM algorithm
is used to solve the resulting system of differential
equations. The advantage of this method lies in the
fact that there would be no need to find the Jaco-
bian matrix and its related decomposed matrix, thus
saving our computational cost and running time. A non-
central 7-point formula was used by Bakodah [111]
to develop numerical scheme, which gives approxi-
mate solution to Burgers’ equation in three different
cases. The solutions were compared using the results
of numerical experiments with 3- and 5-point for-
mulae. The 7-point formula in MOL was used for
solving Burgers’ equation for arbitrary initial condi-
tions.
The purpose of using higher-order discretisation
scheme was ‘the convective term’. MOL was used to
yield non-oscillatory solutions for recirculating flow.
MOL was found to be superior to FDM with respect
to CPU and set-up time. MOL has the simplicity
of the explicit method and stability advantage of the
implicit method. It is possible to achieve higher-order
approximation in the discretisation of spatial deriva-
tive. Due to MOL, comparable orders of accuracy can
be achieved without using extremely small time steps.
Particularly, for more complex and higher Reynold num-
ber problem, MOL is superior to FDM with respect to
CPU time. To decrease the computation time consider-
ably
69 Page 12 of 21 Pramana – J. Phys. (2018) 90:69
we have to increase the order of the spatial
discretisation method (high accuracy with less grid
points),
highly accurate and stable numerical algorithm for
the time integration is needed.
4.6 Finite-element method (FEM) and splines
Finite-element method (FEM) represents a powerful and
general class of techniques for the approximate solution
of PDEs. It gives more accurate solutions compared
to finite-difference methods and can be used in prob-
lems having complicated domain geometry. Another
important method to solve Burgers’ equation is by using
splines. Spline functions have some attractive proper-
ties. Being piecewise polynomial, they can be integrated
and differentiated easily. As they have compact support,
numerical methods in which spline functions are used as
a basis function lead to matrix systems including band
matrices. Such systems have solution algorithms with
low computational cost. Therefore, spline solutions of
the Burgers’ equation are suggested in many studies.
Splines were used in a mathematical context for the first
time by Schoenberg [112] in connection with piecewise
polynomial approximation. Since then, it has come a
long way in solving the Burgers’ equation.
In 1978, Jain and Holla [113] used cubic spline func-
tions for solving coupled Burgers’ equation in two
space variables. They have analysed the algorithms for
their stability and convergence. A finite-element method
based on rectangular elements was developed in [114].
As exact solutions for a two-dimensional Burgers’ equa-
tion were not available, the accuracy of these methods
was checked via grid refinement. In 1980, Varoglu and
Finn [115] applied a finite-element method to solve the
Burgers’ equation, which was based on the combina-
tion of the space–time elements and the characteristics.
Caldwell et al [116] attempted a piecewise approxima-
tion method (finite-element method) using two elements
with the aim of ‘chasing the peak’ by altering the size
of the elements at each stage by using the information at
the previous step. Caldwell and Smith [117] extended
this method to the general case of nelements. They
observed that for large R, finite-element results were
much superior to the finite-difference results. Many
researchers used moving node finite-element method to
solve the Burgers’ equation. Gelinas et al [118]pre-
sented a node moving finite-element method, which can
be applied to large gradients or shocks with high res-
olution and accuracy. In their system, the nodes move
systematically and continuously to those regions where
they are required the most. Caldwell et al [119]fur-
ther developed a moving node finite-element method by
using an algorithm, which was a generalisation of the
one considered by Caldwell et al [116]. Ali and Gard-
ner [120] developed a collocation solution of Burgers’
equation using cubic B-spline finite-element method.
For solving the two-dimensional Burgers’ equations in
inhomogeneous form, a stable scheme based on the
operator-splitting technique with cubic spline functions
was derived by Shankar et al [121]. The uncondition-
ally stable scheme was of first-order accuracy in time
and second-order accuracy in space directions. Özi¸set
al [122] reduced the Burgers’ equation to heat equation
by Hopf–Cole transformation, which was then solved
by FEM. They used linear functions as test functions
over the intervals. But when Hopf–Cole transformation
is used, due to the series solution involved, it would
be required to consider innumerous terms of the series
to ensure fast convergence and minimise error. Dogan
[123] came up with a Galerkin FEM solution of Burgers’
equation using linear elements. The system of ordi-
nary differential equation obtained on applying FEM
was solved by Crank–Nicolson scheme. Many works
on finite-element method have used splines as weight
functions. Kutluay et al [124] developed a least squares
quadratic B-spline FEM. They reduced the Burgers’
equation to a pentadiagonal system by applying clas-
sic weighted residual method over the finite elements,
which was then solved by a variant of Thomas algo-
rithm together with an iteration process at each time step.
Azkan and Ozdes [125] developed a variational method
which was constructed on descretisation in time. The
descretisation was done by replacing the time derivative
with its difference coefficient, thus converting it into an
ordinary differential equation which was then solved by
Galerkin method. Özis et al [126] developed Galerkin
quadratic B-spline finite-element method where the test
function used for FEM was quadratic spline. The ini-
tial interval [0,1] was divided into 80 finite elements of
equal length. The resulting system was first-order ordi-
nary differential equation with pentadiagoanal matrices
as coefficients which was then solved by Thomas algo-
rithm. Aksan [127] brought forward another method
in which Burgers’ equation was converted to a set of
non-linear ordinary differential equations by the method
of discretisation in time and then each of them was
solved by applying the quadratic B-spline finite-element
method. Two test examples established the efficiency
of the presented method. A numerical solution based
on collocation using septic splines was developed by
Ramadan et al [128]. Here, the time descretisation was
done using Crank–Nicolson scheme. The method was
proved to be unconditionally stable. A cubic B-spline
FEM solution of time-splitted Burgers’ equation and
quadratic B-spline FEM solution of space-splitted Burg-
ers’ equation was studied by Da˘get al [129]. Space
splitting was done by setting V=−ux. The first-order
Pramana – J. Phys. (2018) 90:69 Page 13 of 21 69
system thus obtained was solved by quadratic B-spline.
The time-splitted Burgers’ equation was
ut+2uux=0;ut2νuxx =0 (47)
which was then solved using the cubic B-spline. They
found that splitted solution of Burgers’ equation using
splines gave better results compared to non-splitted
solution. Da˘get al [130] further came up with a cubic B-
spline collocation solution of Burgers’ equation, where
the non-linear term uuxwas approximated by using a
form of quasilinearisation as follows:
(uux)n+1
m=un+1
m(ux)n
m+un
m(ux)n+1
m
+un
m(ux)n
m.(48)
Kumar and Mehra [131] proposed a wavelet-Taylor
Galerkin method. In deriving the computational scheme,
Taylor-generalised Euler time discretisation is per-
formed prior to wavelet-based Galerkin spatial approx-
imation. The linear system of equations obtained in
the process is solved by approximate-factorisation-
based simple explicit schemes. Burgers’ equation is also
solved by splitting-up method using a wavelet-Taylor
Galerkin approach. Here, the advection and diffusion
terms in the Burgers’ equation are separated, and the
solution is computed in two phases by appropriate
wavelet-Taylor Galerkin schemes. Ramadan et al [132]
proposed a non-polynomial spline solution of Burgers’
equation. The non-polynomial spline function in this
work has a trigonometric part and a polynomial part of
the first degree. The Cdifferentiability of the trigono-
metric part of non-polynomial spline compensates for
the loss of smoothness inherited by polynomial splines.
Moving boundary conditions were used to improve the
accuracy. In another study of Burgers’ equation by Saka
and Da˘g[133], time and space splitting techniques
were applied to the Burgers’ equation and the modi-
fied Burgers’ equation and then collocation procedure
using quintic B-spline was employed to approximate the
resulting systems. The method was proven to be better
than quadratic B-spline FEM and quartic B-spline col-
location method. Another spline interpolation method
was devised by Jiang and Wang [134]byusingthe
derivative of the quasi-interpolation to approximate the
spatial derivative and a second-order compact finite-
difference scheme to approximate the time derivative.
Two-dimensional Burgers’ equation is solved by local
discontinuous Galerkin (LDG) finite-element method
[135]. The authors have transformed the system of Burg-
ers’ equations to a linear heat equation by means of
Hopf–Cole transformation. Then the LDG method is
used to discretise the heat equation in space. A for-
ward Euler and a third-order Runge–Kutta method is
used to discretise the corresponding ordinary differen-
tial equations. Finally, the numerical solution for the
heat equation is used to obtain numerical solutions of the
system of Burgers’ equations directly. Mittal and Jain
[136] developed a method for solving Burgers’ equation
based on collocation of the modified cubic B-splines
over finite elements. A limiter-free high-order spec-
tral volume formulation was developed to solve the
Burgers’ equation in [137]. They have used Hopf–Cole
transformation to convert non-linear Burgers’ equation
to a linear diffusion equation. The local discontinuous
Galerkin (LDG) and the LDG2 viscous flux discretisa-
tion methods were used to solve this heat conduction
equation. A three-stage SSP Runge–Kutta scheme was
used for time advancement. Numerical solutions are pre-
sented for 1D and 2D Burgers’ equations. Numerical
scheme based on weighted average differential quadra-
ture method was proposed by Jiwari et al [138]. They
have used forward difference method for discretisation
of time variable followed by quasilinearisation tech-
nique and polynomial quadrature method for spatial dis-
cretisation. The resulting linear equations were solved
by Gauss elimination method. Stability and convergence
analysis of the proposed scheme was presented. Yang
[139] presented a finite-volume element method for
approximating the solution of two-dimensional Burg-
ers’ equation. In this paper, Yang used the upwind
technique to handle the non-linear convection term.
Also, the semidiscrete scheme and the fully discrete
scheme are presented. Arora and Singh [140] used
the modified cubic B-splines in differential quadrature
method to give a system of ordinary differential equa-
tion, which was solved by strong stability-preserving
time-stepping Runge–Kutta (SSP-RK43) scheme.
4.7 Other methods
Apart from the methods discussed already, Burgers’
equation has attracted many researchers worldwide to
develop innumerous other methods. Computing the
pressure is the most difficult and CPU consuming part
which requires the solution of a Poisson-type equation
introducing an elliptic nature. Pressure calculation is dif-
ficult and time-consuming leading to the use of pressure
correction methods. The methods are
projection (fractional step) method following Chorin
[141] algorithm,
artificial compressibility method proposed by
Chorin [142],
pressure-based finite volume method (the most well-
known algorithm is the SIMPLE method of Patankar
[143]. Other algorithms are SIMPLER and SIM-
PLEC),
pressure implicit by splitting of operators (PISO)
algorithm by Issa [144].
69 Page 14 of 21 Pramana – J. Phys. (2018) 90:69
Mittal and Singhal [145] gave a spectral method to
solve Burgers’ equation with periodic boundary condi-
tion. They used the fact that the non-linear term uux
is finitely reproducing with respect to basis functions
j=−∞((1/2π)eijx). This method gives a system of
ordinary differential equation which is solved by the
s-stage Runge–Kutta–Chebyshev method. Elton [146]
considered three-lattice Boltzmann methods and an
analogous finite-difference method for solving two-
dimensional viscous Burgers’ equation with periodic
boundary conditions. He proved that in the l1norm, the
lattice Boltzmann methods converge first-order tempo-
rally and second-order spatially. Distributed approxi-
mating functional (DAF) method to Burgers’ equation
for large Reynolds number was developed by Zhang
et al [147]. The method required small number of grid
points and permitted the use of large time steps. Zhang et
al used just about 25–35 grid points to produce highly
accurate solutions when Re 105. The method was
further extended to the case of Re >105by using
200 grid points. They used a large mapping parame-
ter to shift most of the 200 grid points to the boundary
region to obtain oscillation-free solutions. The solu-
tion proved to be much better than those existing in
literature. Park et al [68] devised a method based on
the Karhunen–Loeve decomposition which is a tech-
nique for obtaining empirical eigenfunctions from the
experimental or numerical data of a system. Employing
these empirical eigenfunctions as basis functions of a
Galerkin procedure, one can apriorilimit the function
space considered to the smallest linear subspace that
is sufficient to describe the observed phenomena and
consequently reduce the Burgers’ equation to a set of
ordinary differential equations with a minimum degree
of freedom. Burns et al [148] have considered Burg-
ers’ equation with zero-Neumann boundary conditions
to show that for moderate values of viscosity, numerical
solution approaches non-constant shock-type stationary
solution. Based on Hopf–Cole linearisation, Brander
and Hedenfalk [149] solved Burgers’ equation in one
space dimension for an arbitrary incident pulse of finite
length.
To get accurate results, creation of mesh plays a cru-
cial role in the methods which use discretisation of
PDEs into meshes. But, for discontinuous and high gra-
dient problems mesh generation is a time-consuming
process. Mesh-free or meshless method is a very good
alternative to avoid this trouble. In these methods,
the scattered nodes are only used instead of mesh-
ing the domain of the problem. Hon and Mao [150]
applied a mesh-free method called multiquadratic (MQ)
which is a special type of radial basis function. This
method was originally developed to approximate two-
dimensional geographic surface. Hon and Mao [150]
further developed an adaptive algorithm to adjust MQ
interpolation points to the peak of the shock wave so
as to get better results. Malek and Mansi [151] found a
group theoretic method to solve Burgers’ equation. They
applied one-parameter group transformation to Burgers’
equation along with the initial and boundary values. This
reduced the number of independent variables from two
to one, giving rise to an ordinary differential equation.
The ordinary differential equation is then solved analyt-
ically to get the solution in its closed form. Abbasbandy
and Darvishi [152] studied the Adomian decomposi-
tion method, which decompose the solution θ(x,t)by
an infinite series of components
n=0θn(x,t),where
θnwill be obtained recursively. This gives the solution
as an infinite series usually converging to an accurate
solution. The method has the advantage that it can be
applied directly and does not require any linearisation.
It also does not require descretisation of the variables
and, therefore, it is not affected by errors associated with
discretisation. Abbasbandy and Darvishi [153] applied
modified Adomian’s method (constucted on method
of descretisation in time) to Burgers’ equation. They
used Fourier cosine series approximated to nterms for
θ(x,t0)a0+N
n=1ancos(nπx). The method was
shown to give results better than the FEM results by Özi¸s
et al [122]. A variational iteration method for solving
Burgers’ and coupled Burgers’ equations was developed
by Abdou and Soliman [154], where they constructed a
functional to approximate un+1(x). The accuracy of the
method was established by comparing the results with
those obtained by Adomian decomposition method. In
[155], higher-order accurate two-point compact alter-
nating direction implicit algorithm was developed to
solve the two-dimensional unsteady Burgers’ equation.
The method is the extension of A-stable fourth-order
accurate second-diagonal Pade approximation to solve
multidimensional flow problems. A comparison with
fourth-order Du Fort Frankel scheme is also presented
which is a conditionally stable explicit scheme. Sakai
and Kimura [156] applied two-dimensional Hopf–Cole
transformation to convert Burgers’ equation into lin-
ear heat equation and the resulting equation is solved
by spectral method. A pseudospectral method to solve
Burgers’ equation was explained by Darvishi and Javidi
[157]. This method involves the use of spectral differen-
tiation matrices to find the derivative of u(x)at the col-
location point xj. Runge–Kutta method of fourth order
was used to advance in time. Restrictive Pade approxi-
mation classical implicit finite-difference method was
implemented by Gulsu [158] whose accuracy was
demonstrated by the two test problems. Alice Gorguis
[159] made a comparative study of decomposition
method and Hopf–Cole transformation and established
the advantages such as simplicity and reliability of the
Pramana – J. Phys. (2018) 90:69 Page 15 of 21 69
former over the latter. However, this method demands
the use of truncated series. If the solution series has small
convergence radius, then the truncated series may be
inaccurate in many regions. To enlarge the convergence
domain of the truncated series, Pade approximants (PAs)
to the Adomian’s series solution have been tested and
applied to partial and ordinary differential equations
with good results. Basto et al [160] applied PAs both
in xand tdirections to the truncated series solution
given by Adomian’s decomposition technique for Burg-
ers’ equation. This enlarged the domain of convergence
of the solution and improved the accuracy. In 2008,
Wu and Zhang [161] introduced artificial boundary
method to solve two-dimensional Burgers’ equation in
unbounded domain by means of Hopf–Cole transforma-
tion. Artificial boundaries were introduced to make the
computational domain finite and boundary conditions
on the artificial boundaries were found which reduced
the original problem to an equivalent problem on a
bounded domain. They have also presented the stabil-
ity of the reduced problem. A mesh-free method named
element-free characteristic Galerkin method (EFCGM)
was proposed by Zhang et al [162] for solving Burg-
ers’ equation with various values of viscosity. Based on
the characteristic method, the convection terms of Burg-
ers’ equation disappear and this process makes Burgers’
equation self-adjoint, which ensures that the spatial dis-
cretisation by the Galerkin method can be optimal. The
results were tested for both one-dimensional and two-
dimensional Burgers’ equations. A variational iteration
method was proposed in [163] for solving non-linear
Burgers’ equation in one and two dimensions. Zhang
et al [164] developed another meshless method based
on the coupling between variational multiscale method
and mesh-free methods, for 2D Burgers’ equation.
Zhu et al [165] proposed discrete Adomian decom-
position method (ADM) to numerically solve the two-
dimensional Burgers’ non-linear difference equations.
Asaithambi [166] presented a simple numerical
method based on automatic differentiation or algorith-
mic differentiation that used second-order finite differ-
ences for the spatial derivatives and marched the solution
in time using a Taylor series expansion. The advantage
of automatic differentiation is that it is not necessary
to obtain lengthy algebraic expressions for calculating
higher-order derivatives – they were computed using
recursive formulae obtained from the spatially discre-
tised form of the differential equation itself. Liu and Shi
[167] solved numerically the two-dimensional Burgers’
equations with two variables by the lattice Boltzmann
method. In 2011, Allery et al [168] introduced apriori
reduction method for solving two-dimensional Burgers’
equation. This method is based on an iterative procedure
which consists of building a basis for the solution where
at each iteration the basis is improved. They have shown
that the proposed method takes less computational time.
Another meshless method, the Petrov–Galerkin method,
was presented by Sarboland et al [169]. In this approach,
the trial space was generated by the multiquadratic
radial basis function (MQRBF) and the test space was
generated by the compactly supported RBF. In 2013,
Aminikhah [170] combined Laplace transform and new
homotopy perturbation methods (LTNHPM) to obtain
closed form solutions of the coupled Burgers’ equa-
tion. Aminikhah claimed that the proposed method can
be applied to many complicated linear and non-linear
partial differential equations without doing linearisa-
tion or discretisation. Xie and Li [171] used radial
basis functions to approximate the solution of non-linear
Burgers’ equation. They have used multiquadric radial
basis function for spatial discretisation and a second-
order compact finite-difference scheme for temporal
approximation. A homotopy analysis method (HAM)
based multiscale meshless method was proposed by Mei
[172]. Mei has constructed a multiscale interpolation
operator with radial basis function, while HAM was
used to solve the resulting ODE system. In [173], it was
shown that combining the interval interpolation wavelet
collocation method, HAM-based adaptive precise inte-
gration method can be employed to solve non-linear
Burgers’ equation. A lattice Boltzmann method was
proposed in [174], based on BGK model and Chapman–
Enskoy expansion for computing solutions of Burgers’
equation.
The Lie group method, also called symmetry analy-
sis, is a powerful and direct approach to construct exact
solutions of non-linear differential equations [175]. The
Lie symmetry analysis is performed for the general
Burgers’ equation, which possesses rich solutions and
similarity reductions [176]. Such exact explicit solutions
and similarity reductions are important in both appli-
cations and the theory of non-linear science. Entropic
lattice Boltzmann models are discrete velocity models
of hydrodynamics that possess a Lyapunov function,
which makes them useful as non-linearly stable numer-
ical methods for integrating hydrodynamic equations.
An entropic lattice Boltzmann model for Burgers’ equa-
tion was derived, and used to perform a fully explicit,
unconditionally stable numerical integration of these
equations [177].
4.8 Recent developments
As the Burgers’ equation is non-linear, the numeri-
cal schemes lead to a system of non-linear equations.
Implicit exponential finite-difference method and fully
69 Page 16 of 21 Pramana – J. Phys. (2018) 90:69
implicit exponential finite-difference method for solv-
ing Burgers’ equation was given in [178]. In 2014,
Biazar et al [179] used method of lines to approximate
uxand uxx of one-dimensional quasilinear Burgers’
equation. They discretised spatial derivatives uxand uxx
using first- and second-order central differences. The
resulting system of ordinary differential equation was
solved to arrive at the solution. Another work on Burg-
ers’ equation in 2014 was by Diyer et al [180], where
they gave a numerical scheme based on cubic B-spline
quasi-interpolants and some techniques of matrix argu-
ments. They applied the derivative of the cubic B-spline
quasi-interpolant to approximate the spatial derivative
of the differential equations and employed a first-order
accurate forward difference for approaching the tempo-
ral derivative. So they did not have a system where they
had to invert a matrix but an iterative relationship easy
to implement. Sarboland et al [181], who had devel-
oped mesh-free method in [169] came up with two more
mesh-free methods based on the multiquadric (MQ)
quasi-interpolation operator LW2and direct and indirect
radial basis function network (RBFNs) schemes. Ganaie
and Kukreja [182] in 2014 discussed cubic Hermite col-
location method (CHCM) for Burgers’ equation. They
handled the non-linear term using quasilinearisation.
Time discretisation was done using Crank–Nicolson
scheme. Hermite functions are a class of piecewise
polynomials having continuity properties and play an
important role in setting approximate functions. They
have compact support and can be easily differenti-
ated. Ganaie and Kukreja performed a linear stability
analysis and proved that the method is uncondition-
ally stable. Talwar and Mohanty [183] proposed a new
modified alternating group explicit (MAGE) iterative
method in 2014 for solving one-space-dimensional lin-
ear and non-linear singular parabolic equations. These
methods are explicit in nature and if coupled compactly,
they are suitable for use in parallel computers. Wavelet
solutions of Burgers’ equation with high Reynolds num-
bers were presented by Liu et al [184]. Following
this method, Burgers’ equation was first transformed
into a system of ordinary differential equations using
the modified wavelet Galerkin method. Then, the clas-
sical fourth-order explicit Runge–Kutta method was
employed to solve the resulting system of ordinary dif-
ferential equations. The wavelet algorithm had a much
better accuracy and a much faster convergence rate
than many other numerical methods present in litera-
ture such as finite-difference method, classical weighted
residual method, etc. Gao and Chi [185] proposed a
numerical scheme based on high accuracy multiquadric
quasi-interpolation operator Lw. They have used a mul-
tiquadric quasi-interpolant to approximate derivatives of
the solution in spatial domain. Finite difference was used
to approximate the derivatives of solution in temporal
domain. An all-at-once approach for the optimal control
of the unsteady Burgers’ equation is given by Yilmaz
and Karaso¨zen [186]. They have discretised the non-
linear Burgers’ equation in time using the semi-implicit
discretisation and the resulting first-order optimality
conditions are solved iteratively by Newton’s method.
An explicit solution of Burgers’ equation with station-
ary point source is given in [187]. This paper also
explains the role of diffusion and convection when
a non-autonomous reaction term produces heat con-
stantly. In [188], a local RBF-based method of approxi-
mate particular solutions for two-dimensional unsteady
Burgers’ equations is developed. A comparative study
between the lattice Boltzmann method (LBM) and
the alternating direction implicit (ADI) method is pre-
sented in [189] using the 2D steady Burgers’ equation.
The comparative study showed that the LBM performs
comparatively poor on high-resolution meshes due to
smaller time step sizes, while on coarser meshes where
the time step size is similar for both methods, the cache-
optimized LBM performance is superior. In [190], an
implicit logarithmic finite-difference method is intro-
duced for the numerical solution of one-dimensional
coupled non-linear Burgers’ equation. The numerical
scheme provides a system of non-linear difference equa-
tions which are linearised using Newton’s method. The
obtained linear system is solved by Gauss elimination
with partial pivoting algorithm. Goyal and Mehra [191]
developed a fast adaptive diffusion wavelet method for
solving 1D and 2D coupled Burgers’ equation with
Dirichlet and periodic boundary conditions. They con-
structed diffusion wavelet from the diffusion operator
obtained by discretising the Burgers’ equation. They
have used diffusion wavelet for the construction of an
adaptive grid as well as for the computations involved in
the numerical solution of Burgers’ equation. Moreover,
by comparing the CPU time, they claimed that the pro-
posed method is faster. Laplace decomposition method
(LDM) is proposed to solve the two-dimensional non-
linear Burgers’ equations in [192]. Convergence and
accuracy of the proposed scheme are shown through test
problems. Kumar and Pandit [193] introduced a numeri-
cal scheme based on finite difference and Haar wavelets
to solve coupled Burgers’ equation. They have discre-
tised the time derivative by forward difference, followed
by quasilinearisation technique to linearise the coupled
Burgers’ equation. Space derivatives are discretised with
Haar wavelets resulting in a system of linear equations
whichissolvedusingMatlab7.0.
After Hopf and Cole introduced the transformation,
several attempts have been made to generalise Cole–
Hopf transformation, which were nicely documented
in [194]. For solving the two-dimensional non-linear
Pramana – J. Phys. (2018) 90:69 Page 17 of 21 69
Burgers’ equation, a numerical scheme based on high
accuracy MQ quasi-interpolation scheme was presented
by Sarboland et al [195] in 2015. They used equidis-
tant data in numerical experiments. Yet again in 2015,
Mukundan and Awasthi [196] presented new and effi-
cient numerical techniques for solving Burgers’ equa-
tion. They used Hopf–Cole transformation to get one-
dimensional diffusion equation which was semidiscre-
tised by using method of lines (MOL). Resulting system
of ODEs was solved by backward differentiation formu-
lae (BDF) of order one, two and three. Amit Prakash et
al [197] used the fractional variational iteration method
(FVIM) to solve a time- and space-fractional coupled
Burgers’ equations. The results obtained by FVIM were
more accurate than FVIM, ADM, GDTM and HPM
methods. In 2015, Saleeby [198] characterised the mero-
morphic solution of generalised Burgers’ equation given
by ux+umuy=0, where m0 is an integer. In the
same year, Jiwari [199] developed a hybrid numerical
scheme based on forward finite difference, quasilineari-
sation and uniform Haar wavelets. In this paper, the non-
linear Burgers’ equation was discretised along temporal
direction by means of Euler implicit method. It was fol-
lowed by the quasilinearisation technique in order to lin-
earise the stationary Burgers’ equation. Finally, uniform
Haar wavelets were used for spatial discretisation. Jiwari
showed that the proposed scheme provides better accu-
racy than other existing numerical techniques. More-
over, the proposed scheme can capture the behaviour
of numerical solution for small values of kinematic vis-
cosity, ν, and hence overcome the drawback of their
previous paper [200]. Higher-order accurate numerical
solution for one version of two-dimensional unsteady
Burgers’ equation was proposed by Zhanlav et al [201]
in 2015. They used the linear transformation z=x+y,
s=xyand ¯
t=2tto reduce 2D Burgers’ equation
ut+u(ux+uy)
=ν(uxx +uyy), (x,y)∈[a,b]X[c,d]
into
ut+uuz=ν(uzz +uss), a+czb+d.(49)
If solution u(z,s,¯
t)depends only on sand ¯
tvari-
ables, eq. (49) reduces to heat equation. On the other
hand, if solution u(z,s,¯
t)depends only on zand ¯
t
variables, eq. (49) reduces to one-dimensional Burg-
ers’ equation. They have used Cole–Hopf transfor-
mation to reduce 1D Burgers’ equation into heat
equation. Finally, heat equation with Robin boundary
conditions is solved by a three-level explicit finite-
difference scheme. This scheme is sixth-order accurate
in space and third-order accurate in the time variable.
Numerical solution of heat equation was used to find
numerical solution of 1D Burgers’ equation which in
turn was used to find numerical solution of 2D unsteady
Burgers’ equation. Zhanlav et al [201] claimed that
adoption of this method reduces the computational
costs compared to other direct methods for solving
the 2D unsteady Burgers’ equation. Numerical scheme
for the coupled Burgers’ equation based on colloca-
tion of the modified bi-cubic B-spline functions was
proposed by Mittal and Tripathi [202]. They have
used these functions for space variables and for their
derivatives. The resulting system of first-order ordinary
differential equations was solved by strong stability pre-
serving Runge–Kutta method (SSP-RK54). They have
compared the obtained numerical results with those
suggested in earlier studies. Mittal et al [203]pro-
posed a numerical method based on the properties
of uniform Haar wavelets together with a collocation
method and semidiscretisation along the space direc-
tion for solving a coupled viscous Burgers’ equation.
The semidiscretisation scheme forms a system of non-
linear ordinary differential equations which is solved
by the fourth-order Runge–Kutta method. Mohanty
et al [204] presented a new two-level implicit com-
pact operator method for the numerical simulation of
coupled viscous Burgers’ equation in one spatial dimen-
sion. This scheme has accuracy of order two in time
and four in space. Mohanty et al have used three spa-
tial grid points and the obtained non-linear system was
solved by Newton’s iterative method. Lie group method
was used for the analysis of the generalised system of
2D Burgers’ equations with infinite Reynolds number
[205]. Abdulwahhab [205] has derived optimal sys-
tem of one-dimensional subalgebras which was further
used to obtain generalised distinct exact solutions of the
velocity components.
Recently in 2016, two new modified fourth-order
exponential time differencing Runge–Kutta (ETDRK)
schemes in combination with a global fourth-order
compact finite-difference scheme (in space) for direct
integration of non-linear coupled viscous Burgers’ equa-
tions is presented in [206]. One of the scheme is a
modification of the Cox and Matthews ETDRK4 scheme
based on (1,3)-Padé approximation and the other is a
modification of Krogstads ETDRK4-B scheme based
on (2,2)-Padé approximation. Bhatt and Khaliq [206]
have presented and compared the accuracy and effi-
ciency of the proposed modified schemes. Bonkile et
al [207] sketched a new implicit scheme with second-
order accuracy in space and time, which was proposed to
solve Burgers’ equation without using Hopf–Cole trans-
formation. Seydao˘glu et al [208] proposed higher-order
splitting methods with complex coefficients and extrap-
olation methods for treating one-dimensional Burgers’
equation. As splitting methods with real coefficients
of order higher than two involve negative time steps,
69 Page 18 of 21 Pramana – J. Phys. (2018) 90:69
it is not suitable for time-irreversible systems such
as Burgers’ equation. Hence, the splitting technique
with complex coefficients is used for solving Burgers’
equation with periodic, Dirichlet, Neumann and Robin
boundary conditions. A high-order finite-volume com-
pact scheme is introduced by Guo et al [209]tosolve
one-dimensional Burgers’ equation. They have com-
puted the non-linear advective terms by the fifth-order
finite-volume weighted upwind compact scheme. The
diffusive terms are discretised by using the finite-volume
six-order Padé scheme and the strong stability preserv-
ing third-order Runge–Kutta time discretisation is used
in this work.
Two-dimensional Burgers’ equation is given by
ut+u(ux+uy)=ν(uxx +uyy), (50)
vt+u(ux+uy)=ν(vxx +vyy)(51)
subject to initial condition
u(x,y,0)=f(x,y), (x,y)D,(52)
v(x,y,0)=g(x,y), (x,y)D(53)
and boundary conditions
u(x,y,t)=f1(x,y,t), (x,y)D,t>0,(54)
v(x,y,t)=g1(x,y,t), (x,y)D,t>0,(55)
where Dis the domain, Dis its boundary, u(x,y,t)
and v(x,y,t)are the velocity components to be deter-
mined, f,g,f1and g1are known functions and νis
the kinematic viscosity parameter. Sinuvasan et al [210]
introduced an inhomogeneous term, the function f(t,x)
in Burgers’ equation, such that there exists at least one
symmetry for the whole equation. For the special cases,
the given equation was reduced to the equation for a
linear oscillator with non-constant coefficient.
5. Conclusions
Advances in computational capacities and new robust
schemes will encourage multidisciplinary researchers
to consider Burgers’ equation more enthusiastically.
Meanwhile, availability of exact solution gives an extra
advantage to Burgers’ equation. The main objective of
this paper is to review a brief history from physical and
mathematical point of view and recent developments in
numerical simulation of Burgers’ equation from simple
schemes to the most efficient one. Additionally, a sys-
tematic categorisation of large scientific data available
in the literature due to the tremendous achievements of
various research groups is done. Researchers are in con-
stant search of more accurate, stable, robust and efficient
scheme, which will lead to the development of signifi-
cantly improved schemes. In this detailed discussion,
we sincerely emphasised the importance of Burgers’
equation in modern engineering scenario. We would like
to predict a bright future ahead.
Acknowledgements
The authors thank the anonymous referees for their valu-
able time, effort and extensive comments which help to
improve the quality of this paper.
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... The inverse Péclet number ǫ appearing in Eq. (12) plays the role of viscosity in the more common form of the Burgers equation, which was initially introduced in the context of fluid turbulence [56,57]. The effect of evaporation is embodied through an extra linear term monitored by the Damköhler number α. ...
... The situation most amenable to analytical treatment is when diffusion and evaporation are both absent (α = ǫ = 0 or Da = Pe −1 = 0). The equation to solve in this case is the inviscid Burgers equation [56,57,59] ...
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When surface-active molecules are released at a liquid interface, their spreading dynamics is controlled by Marangoni flows. Though such Marangoni spreading was investigated in different limits, exact solutions remain very few. Here we consider the spreading of an insoluble surfactant along the interface of a deep fluid layer. For two-dimensional Stokes flows, it was recently shown that the nonlinear transport problem can be exactly mapped to a complex Burgers equation [D. Crowdy, SIAM J. Appl. Math. 81, 2526 (2021)]. We first present a very simple derivation of this equation. We then provide fully explicit solutions and find that varying the initial surfactant distribution—pulse, hole, or periodic—results in distinct spreading behaviors. By obtaining the fundamental solution, we also discuss the influence of surface diffusion. We identify situations where spreading can be described as an effective diffusion process but observe that this approximation is not generally valid. Finally, the case of a three-dimensional flow with axial symmetry is briefly considered. Our findings should provide reference solutions for Marangoni spreading that may be tested experimentally with fluorescent or photoswitchable surfactants.
... The inverse Péclet number ǫ appearing in Eq. (12) plays the role of viscosity in the more common form of the Burgers equation, which was initially introduced in the context of fluid turbulence [56,57]. The effect of evaporation is embodied through an extra linear term monitored by the Damköhler number α. ...
... The situation most amenable to analytical treatment is when diffusion and evaporation are both absent (α = ǫ = 0 or Da = Pe −1 = 0). The equation to solve in this case is the inviscid Burgers equation [56,57,59] ...
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Full-text available
When surface-active molecules are released at a liquid interface, their spreading dynamics is controlled by Marangoni flows. Though such Marangoni spreading was investigated in different limits, exact solutions remain very few. Here we consider the spreading of an insoluble surfactant along the interface of a deep fluid layer. For two-dimensional Stokes flows, it was recently shown that the non-linear transport problem can be exactly mapped to a complex Burgers equation [Crowdy, SIAM J. Appl. Math. 81, 2526 (2021)]. We first present a very simple derivation of this equation. We then provide fully explicit solutions and find that varying the initial surfactant distribution - pulse, hole, or periodic - results in distinct spreading behaviors. By obtaining the fundamental solution, we also discuss the influence of surface diffusion. We identify situations where spreading can be described as an effective diffusion process but observe that this approximation is not generally valid. Finally, the case of a three-dimensional flow with axial symmetry is briefly considered. Our findings should provide reference solutions for Marangoni spreading, that may be tested experimentally with fluorescent or photoswitchable surfactants.
... Some nonlinear partial differential equations with nonlinear convection terms uu x were studied by Rasoulizadeh et al. [30,31], Qiu et al. [28] and Zhang et al. [42]. Furthermore, readers can find more details about this subject in the review paper [5] and its references. ...
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This paper formulates a third-order backward differentiation formula (BDF3) fourth-order compact difference scheme based on a developed fourth-order operator for computing the approximate solution of the Burger's equation. The equation is one of the useful description for modeling nonlinear acoustics, gas dynamics, fluid mechanics and etc. This proposed approach approximates the solution of Burger's equation with the help of two main steps. In the first step, the temporal discretization is accomplished by virtue of the BDF3 approach. In the second step, a developed fourth-order operator and the classic compact difference formula combine with the method of order reduction are applied for spatial discretization, thereby constructing a fully-discrete scheme. The theoretical analysis is proved in detail by means of the discrete energy method. The proposed scheme is convergent with third order for time and fourth order for space. Numerical results are carried out to verify the validity and accuracy of the proposed method.
... The Burgers equation is one of the most popular nonlinear one dimensional wave equations whose analytical and numerical solution is well known and documented in the literature. For a comprehensive historical overview, the reader is referred to [15] and [16] and the references contained there. Normally, the dimensionless form of the Burgers equation (see, for example, [17, equation (2.76)]) is more used than the above equation. ...
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In this paper, we present a novel and simple Yee Finite-Difference Time-Domain (FDTD) scheme to solve numerically the nonlinear second-order thermoviscous Navier–Stokes and the Continuity equations. In their original form, these equations can not be discretized by using the Yee’s mesh, at least, easily. As it is known, the use of the Yee’s mesh is recommended because it is optimized in order to obtain higher computational performance and remains at the core of many current acoustic FDTD softwares. In order to use the Yee’s mesh, we propose to rewrite the aforementioned equations in a novel form. To achieve this, we will use the substitution corollary. This procedure is novel in the literature. Although the scheme can be extended to more than one dimension, in this paper, we will focus only on the one-dimensional solution because it can be validated with two analytical solutions to the Burgers equation: the Mendousse mono-frequency solution and the Lardner bi-frequency solution. Numerical solutions are excellently consistent with the analytical solution, which demonstrates the effectiveness of our formulation.
... Fractional order diffusion equations describe anomalous diffusion phenomena, which aid in the analysis of systems such as: plasma diffusion, fractal diffusion, anomalous diffusion on liquid surfaces, analysis of heart beat histograms in healthy individuals, among other physical systems (see e.g. (BILER;FUNAKI;WOYCZYNSKI, 1998) and (BONKILE et al., 2018)). In order to understand the following text, we need the following definition which can be found in (KILBAS; SRIVASTAVA; TRUJILLO, 2006). ...
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In this work, we develop the Yamazaki inequality and estimates of the Mittag-Leffler family associated with the space-time fractional diffusion equation in 𝐿𝑝-weak spaces. This result generalizes the inequality given by Yamazaki (2000) for the heat semigroup, which was generalized by Ferreira and Villamizar-Roa (2006) for heat semigroup associated with the fractional Laplacian, and Caicedo et al (2021) for the Mittag-Leffler family associated with the time-fractional diffusion equation.
... The v iscous Burgers equation is a fundamental second-order semilinear PDE which is frequently employed to model physical phenomena in fluid dynamics [48] and nonlinear acoustics in dissipative media [49]. Its general form is ...