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mathematics

Article

A Meshless Method for Burgers’ Equation Using

Multiquadric Radial Basis Functions With

a Lie-Group Integrator

Muaz Seydao˘glu

Department of Mathematics, Faculty of Art and Science, Mu¸s Alparslan University, 49100 Mu¸s, Turkey;

m.seydaoglu@alparslan.edu.tr

Received: 27 November 2018; Accepted: 21 January 2019; Published: 22 January 2019

Abstract:

An efﬁcient technique is proposed to solve the one-dimensional Burgers’ equation based on

multiquadric radial basis function (MQ-RBF) for space approximation and a Lie-Group scheme for

time integration. The comparisons of the numerical results obtained for different values of kinematic

viscosity are made with the exact solutions and the reported results to demonstrate the efﬁciency and

accuracy of the algorithm. It is shown that the numerical solutions concur with existing results and

the proposed algorithm is efﬁcient and can be easily implemented.

Keywords:

Burgers’ equation; meshless method; multiquadric radial basis function (MQ-RBF);

Lie-group method

1. Introduction

We consider the nonlinear Burgers’ equation in the one-dimensional case

∂u(x,t)

∂t+∂

∂xF(u(x,t) = v∂2u(x,t)

∂x2(1)

with initial condition

u(x, 0)=u0(x),a≤x≤b

and boundary conditions

u(a,t)=h1(t),u(b,t)=h2(t), 0 ≤t(2)

where,

F(u(x,t)) =1

2u(x,t)2

,

v>

0 is interpreted as kinematic viscosity number and

u0(x)

,

h1(t)

,

h2(t)

are known sufﬁciently smooth functions of the time and space variables, respectively. The steady

solutions of the Burgers’ equation, for the ﬁrst time, are presented by Bateman [

1

]. Burger [

2

] proposed

this equation as a model of turbulence. The Burgers equation is considered to be a simpliﬁed model of

the Navier-Stokes equation because they share common nonlinear and viscosity terms. Additionally,

it may be helpful to understand the physical mechanisms of the wave motion from advection and

dissipation terms. The analytical solutions in the terms of inﬁnite series can be computed by using the

Hopf-Cole transformation [

3

,

4

] and this gives the opportunity for a comparison between numerical

algorithms. Throughout history, there have been many investigations of Burgers’ equation in a variety

of different areas including nonlinear wave, shock waves, trafﬁc ﬂows, gas dynamics, elasticity, etc.

(see [

5

] and references therein). For all these reasons, Burgers’ equation has been dealt with by many

researchers. The authors of [

6

] examined different exact solutions of the one-dimensional Burgers’

equation. Many researchers approximate the solutions of the Burgers’ equation by various numerical

Mathematics 2019,7, 113; doi:10.3390/math7020113 www.mdpi.com/journal/mathematics

Mathematics 2019,7, 113 2 of 11

schemes such as ﬁnite difference, ﬁnite element, exponentially ﬁtted, Haar wavelet, differential

quadrature [7–11].

A kind of univariate multiquadric (MQ) quasi-interpolation technique to approximate a solution

of the Burgers’ equation is tailored by Chen and Zu [

12

]. A meshless method of lines based on

radial basis functions (RBFs) is proposed for the numerical solution of Burgers’-type equations in [

13

].

Xie et al. [14]

presented the method of particular solutions based on RBFs and ﬁnite difference scheme

to solve one-dimensional time-dependent inhomogeneous Burgers’ equations. Two mesh-free methods,

in which the MQ quasi-interpolation method is applied in direct and indirect forms for the numerical

solution of the Burgers’ equation, are proposed in

[15]. Fan et al. [16]

developed a mesh-free numerical

method based on a combination of the multiquadric RBFs (LRBFCM) and the ﬁctitious time integration

method (FTIM) to solve the two-dimensional Burgers’ equations. Bouhamidi et al. [

17

] presented

RBFs interpolation technique for spatial discretization and implicit Runge–Kutta (IRK) schemes for

temporal discretization of the unsteady coupled Burgers’-type equations. Xie et al. [

18

] approximated

the solution of the Burgers’ equation spatially by the multiquadric MQ-RBF and used C-N ﬁnite

difference scheme as temporal discretization technique.

On the other hand, the goal of Geometric Numerical Integration (GNI) is to preserve geometric

structure of the systems when approximating their solutions. Thus, the discretized solution of the

differential equation has the same qualitative properties of the system [

19

,

20

]. Lie-group methods are

a class of geometric numerical integrators which preserve qualititative structure of the exact solution.

Thus, one can preserve the structural properties of the true ﬂows [

21

]. The authors of [

22

] presented

a group preserving scheme based on Cayley transformation for Burgers’ equation, in which ﬁnite

difference method is used for spatial discretization. Furthermore, a combination of the one-step

backward group preserving scheme (BGPS) and the numerical method of line to discretize the time

and spatial variables, respectively, is presented in [23] to solve Burgers’ equation numerically.

In the present work, the numerical solutions of the one-dimensional Burgers’ equation is analyzed

by the MQ-RBFs and a Lie-Group version of Euler method for space and temporal discretization

technique, respectively. The goal is to take advantage of both meshless method and GNI method for

numerical approximations obtained by this combination of the Lie-Group method based on radial

basis functions (LG-RBFs). This method has been recently applied to the Heat equation [

24

] and

high-dimensional generalized Benjamin-Bona-Mahony-Burgers’ equation [25] .

2. Numerical Scheme

In this section, we introduce a scheme which consists of the MQ-RBFs and the first-order explicit

Lie-group version of Euler method to solve Burgers’ equation numerically.

2.1. The Multiquadric Radial Basis Approximation

We approximate Burgers’ equation spatially by the MQ-RBFs to obtain a nonlinear system of

ordinary differential equations. The approximation of the function u(x,t) can be written as

u(x,t)≈

M

∑

j=1

λj(t)ϕrj+λM+1(t)x+λM+2(t), (3)

where the MQ-RBFs deﬁned as

ϕ(x) = ϕrj=qr2

j+c2, (4)

where

rj=

x−xj

is the Euclidean norm and the unknown functions

λj(t)

’s can be determined

by the collocation technique [

26

,

27

]. The number

c

appeared in Formula

(4)

interpreted as a shape

parameter and it plays an important role in the accuracy of the schemes. One can control the shape

of RBFs with the free parameter

c

. Thus, the selection of its appropriate value is very important to

minimize the error of approximation. Mostly authors use "trial and error" to describe an acceptable

Mathematics 2019,7, 113 3 of 11

value of shape parameter for highly accurate results. We also follow the same technique to compute

shape parameters for experiments. However, collocating

(3)

at the M points yields the M equations,

an extra two equations are required. This is assured by considering additional two conditions for

(3)

given as [27]

M

∑

j=1

λjn=

M

∑

j=1

λjnxj=0, (5)

where λjn=λj(tn). Following [25], one can write

ϕrij =q(xi−xj)2+c2,

where the collocation points

{xi}

on the interval [a, b] are called for 2

≤i≤M−

2 as center points and

for

i=

1 and

i=M

as boundary points. If one inserts Equation

(3)

for 2

≤i≤M−

2 into Equation

(1)

then obtains following nonlinear system of differential equations

d

dt u(t)=f(t,u(t)), (6)

where

u(t)=M

∑

j=1

λj(t)ϕrij +λM+1(t)+λM+2(t)i=M−1

i=2

, (7)

and

f(t,u(t)) = ν

M

∑

j=1

λj(t)∂2ϕrij

∂x2(8)

−∂

∂xF(

M

∑

j=1

λj(t)ϕrij +λM+1(t)x+λM+2(t))i=M−1

i=2

.

2.2. Lie-Group Method

We have used Lie-group version of Euler method as a temporal approximation scheme to solve

the system of ordinary differential Equation (6) numerically. Using Equation (6) one has

d

dt kuk=f.u

kuk. (9)

Merging Equations (6) and (9) leads to a system of differential equations

dt

dU=AU, (10)

where

U="u

kuk#,

is an augmented vector,

A=

0d×df(t,u)

kuk

fT(t,u)

kuk0

, (11)

Mathematics 2019,7, 113 4 of 11

is an augmented matrix of state variables [

28

]. Notice that the ﬁrst equation in

(10)

is the same as

Equation

(6)

and the second one is known as Minkowskian structure for the augmented system.

The augmented vector U veriﬁes the cone condition

UTgU =u.u−kuk2=kuk2−kuk2=0, (12)

where Minkowski metric gis deﬁned as

g="Id×d0d×1

01×d−1#,

d×d

identity matrix given with

Id×d

and the dot between two

d

-dimensional vectors deﬁnes the

Euclidean inner product [

28

]. Furthermore, the matrix

A

deﬁned by (11) belongs to the Lie algebra of

the Lorentz group SO0(d, 1), i.e.,

ATg+gA =0.

Now we have extended

(d+

1

)

-dimensional system (10) of the

d

-dimensional system (6), which has

the additional property that the solutions should satisfy cone condition (12). Thus, it is very important

to preserve this property by the numerical integrators. All one must do is to check the properties

GTgG =g,

detG =1,

G0

0>0,

with G(tn)∈SO0(d, 1)

Un+1=G(tn)Un, (13)

where

Un≈U(tn)

and

G0

0

is the zeroth component of

G

[

28

]. A Lie-group version of ﬁrst-order Euler

method can be applied to solve Equation (10) as follows

Un+1=exp(∆tA(tn))Un,

where Un+1∈SO0(d, 1). On the other hand, it is easy to express exp(∆tA(tn)) in closed-form

exp(∆tA(tn)) =

In+(an−1)

kfnk2fnfT

n

bnfn

kfnk

bnfT

n

kfnkan

,

where , un≈u(tn),∆t=tn+1−tn,fn=f(un,tn)and

an=cosh ∆tkfnk

kunk,bn=sinh ∆tkfnk

kunk.

Inserting this expression of exp(∆tA(tn)) for G(tn)into (13) gives

un+1=un+ζnfn, (14)

ζn=(an−1)fn.un+bnkunk k fnk

kfnk2(15)

kun+1k=bn(fn.un) + ankunk k fnk

kfnk. (16)

Mathematics 2019,7, 113 5 of 11

The Equations (14)–(16) preserve the cone condition for each advanced time. Furthermore,

the scheme (14) unconditionally preserves the ﬁxed point and the geometric property of the true ﬂows

of the original equation (see [25], Theorem 1,2).

Now we apply scheme (14) to solve Equation (6) given with Equations (7)–(8). Let

tn=n∆t,n=0, . . . , N

be discrete time points with steplength

∆t=T/N

. If one considers scheme (14) for

n=

0 at initial

time t0=0 then has

M

∑

j=1

λ1

jϕrij +λ1

M+1xi+λ1

M+2=u0(xi) + ζ0f0,i=2, . . . , M−1,

where

f0=hu00

0(xi)−F0(u0(xi))iM−1

i=2

and

ζ0

can be computed from Equation (15). For 1

≤n≤N−

1,

i=2, . . . , M−1 one has

M

∑

j=1

λn+1

jϕrij +λn+1

M+1xi+λn+1

M+2=

M

∑

j=1

λn

jϕrij +λn

M+1xi+λn

M+2+ζnfn, (17)

where

fn="v

M

∑

j=1

λn

j

∂2ϕrij

∂x2−∂

∂xF(

M

∑

j=1

λn

jϕrij +λn

M+1x+λn

M+2)#i=M−1

i=2

, (18)

and

ζn

can be computed from Equation (15). One can write Equations (5), (17), (18) and boundary

conditions (2) in the matrix form as follows

Dλn+1=Bn+1(19)

where

D=

ϕ(r11)ϕ(r12 ). . . ϕ(r1M)x11

ϕ(r21)ϕ(r22 ). . . ϕ(r2M)x21

.

.

..

.

.....

.

..

.

..

.

.

ϕ(rM1)ϕ(rM2). . . ϕ(rMM )xM1

x1x2. . . xM0 0

1 1 . . . 1 0 0

,

λn+1=hλn+1

1,λn+1

2, . . . , λn+1

M,λn+1

M+1,λn+1

M+2iT,

and

Bn+1=hbn+1

1,bn+1

2, . . . , bn+1

M, 0, 0iT.

Each elements of vector Bcan be computed for i=2, . . . , M−1 as

b1

i=u0(xi) + ζ0f0,

bn+1

i=

M

∑

j=1

λn

jϕrij +λn

M+1xi+λn

M+2+ζnfn,n=1, . . . , N−1,

and for

i=

1,

M

as

bn+1

1=hn+1

1

,

bn+1

M=hn+1

2

, respectively. The coefﬁcients

λn+1

j

are computed by

solving the resulting system of (19) and then the numerical solutions are obtained using them in (3).

The matrix

D

can be shown to be invertible and it is often ill-conditioned. One can determine how

Mathematics 2019,7, 113 6 of 11

accurately to solve system of (19) by considering the conditioning of

D

. The condition number of this

system related to small perturbation in Dis deﬁned for any norm k.kas

κ(D) = kDk

D−1

.

It is possible to lose

log10κ(D)

digits through the computational process of the system (19) including

ill-conditioned matrix

D

[

29

]. To solve ill-posed system (19) we have used Gaussian elimination with

partial pivoting. We obtained the solutions by using the standard ﬂoating-point arithmetic in our

computational algorithm. The shape parameter also plays an important role for the conditioning of the

matrix

D

. One can have high accuracy with smaller value of

c

for a ﬁxed number of points

M

and in

this case one often faces with the ill-conditioned matrix

D

. On the other hand, the condition number

κ(D)increases with the number of collocation points Mfor ﬁxed the values of c.

One can investigate the numerical stability of the method of line by Rule of Thumb [

30

]. If the

eigenvalues of the discretized spatial operator, scaled by steplength ∆t, are inside the stability region

of Lie-Group version of Euler integrator than the proposed algorithm is convergent.

3. Numerical Results

We consider Burgers’ equation on the interval [0, 1]with the following initial condition

u(x, 0) = sin(πx),

and the following boundary conditions

u(0, t) = u(1, t) = 0, t>0.

Its analytical solution is given with the Hopf-Cole transformation as [3,4]

u(x,t) = 2νπ ∑∞

i=1kiexp(−i2π2νt)isin(iπx)

k0+∑∞

i=1kiexp(−i2π2νt)cos(iπx), (20)

where

k0=Z1

0exp n−(2πν)−1[1−cos(πx)]odx,

ki=2Z1

0exp n−(2πν)−1[1−cos(πx)]ocos(iπx)dx,

(i=1, 2, 3 . . .).

We compute the analytical solutions from (20) up to 500 series terms for all comparisons in

experiments. The shape parameters are computed by trial and error. We compare the errors for

different values of shape parameters which are leading the lower conditioned number of matrix

D

in (19). We select the shape parameter c for the lowest error values in each experiment. Accuracy of

these methods is examined by computing error in L2and L∞norm.

uapr −u

L2= h

M

∑

j=1

(uj,N−u(xj,t))2!1/2

,

uapr −u

L∞=max

1≤j≤Muj,N−u(xj,t).

We compute the results of the LG-RBFs for

ν=

1 at different ﬁnal times

t=

0.1, 0.15, 0.2, 0.25

with respectively

c=

0.299, 0.297, 0.299, 0.299 and at points

x=

0.25, 0.5, 0.75. We compare the

numerical solutions of the LG-RBFs with GPS [

22

] for

M=

40,

∆t=

10

−4

and with [

31

] for

M=

80,

Mathematics 2019,7, 113 7 of 11

∆t=

10

−5

.The all results tabulated in Table 1demonstrate that the LG-RBFs method is highly accurate

compared to that of other methods. Now we compute the solutions with LG-RBFs for

ν=

0.1

at different times

t=

0.4, 0.6, 0.8, 1, 3 with respectively

c=

0.128, 0.312, 0.312, 0.312 and at points

x=

0.25, 0.5, 0.75. Numerical solutions obtained by LG-RBFs with

M=

40,

∆t=

10

−3

, have been

tabulated and compared in Table 2with exact solution and the numerical solutions presented in [

22

]

for

M=

40,

∆t=

10

−3

and [

31

] for

M=

80,

∆t=

10

−4

. It is clear from table that LG-RBFs provides

accurate results than are obtained by [

22

,

31

]. Finally, we compute the solutions with LG-RBFs for

ν=

0.01 at different times

t=

0.4, 0.6, 0.8, 1, 3 with respectively

c=

0.121, 0.034, 0.036, 0.114, 0.111 and

at points

x=

0.25, 0.5, 0.75. The results compared in Table 3with exact solutions and the numerical

solutions presented in [

7

,

22

,

31

] with

M=

80,

∆t=

10

−4

for all methods. Table 3indicate that the

accuracy of LG-RBFs solutions is compatible with the numerical solutions obtained in [

7

,

22

,

31

]. It is

easy to see that the all results tabulated in Tables for the LG-RBFs are in good agreement with the

analytical solutions.

Table 1.

Comparison of the approximate solution at different final times with

v=

1,

M=

40 and

∆t=

0.0001.

x t Galerkin [22] GPS [22] LG-RBFs Exact

0.25 0.10 0.25469 0.25376 0.25364 0.25364

0.15 0.15672 0.15672 0.15660 0.15660

0.20 0.09619 0.09654 0.09644 0.09644

0.25 0.05924 0.05929 0.05922 0.05922

0.50 0.10 0.37134 0.37177 0.37157 0.37158

0.15 0.22674 0.22700 0.22682 0.22682

0.20 0.13829 0.13862 0.13847 0.13847

0.25 0.08457 0.08464 0.08454 0.08454

0.75 0.10 0.27102 0.27273 0.27258 0.27258

0.15 0.16411 0.16450 0.16437 0.16437

0.20 0.09929 0.09954 0.09943 0.09944

0.25 0.06036 0.06042 0.06035 0.06035

Table 2.

Comparison of the approximate solutions at different final times with

v=

0.1,

M=

40 and

∆t=

0.001.

x t Galerkin [22] GPS [22] LG-RBFs Exact

0.25 0.4 0.31429 0.30889 0.30864 0.30889

0.6 0.24373 0.24077 0.24061 0.24074

0.8 0.19758 0.19573 0.19559 0.19568

1.0 0.16391 0.16264 0.16251 0.16256

3.0 0.02743 0.02725 0.02720 0.02720

0.50 0.4 0.57636 0.56988 0.56944 0.56963

0.6 0.45169 0.44745 0.44707 0.44721

0.8 0.36245 0.35948 0.35914 0.35924

1.0 0.29437 0.29215 0.29184 0.29192

3.0 0.04057 0.04028 0.04020 0.04020

0.75 0.4 0.62952 0.62605 0.62564 0.62544

0.6 0.49034 0.48778 0.48727 0.48721

0.8 0.37713 0.37438 0.37389 0.37392

1.0 0.29016 0.28784 0.28741 0.28747

3.0 0.01334 0.02983 0.02977 0.02977

Mathematics 2019,7, 113 8 of 11

Table 3.

Comparison of the approximate solutions at different ﬁnal times with

v=

0.01,

M=

80 and

∆t=0.0001.

x t Exact-Explicit [7] RK4 [22] GPS [22] LG-RBFs Exact

0.25 0.4 0.34164 0.34197 0.34193 0.34189 0.34191

0.6 0.26890 0.26900 0.26897 0.26890 0.26896

0.8 0.22150 0.22151 0.22149 0.22140 0.22148

1.0 0.18825 0.18821 0.18820 0.18816 0.18819

3.0 0.07515 0.07512 0.07511 0.07512 0.07511

0.50 0.4 0.65606 0.66083 0.66079 0.66068 0.66071

0.6 0.52658 0.52950 0.52946 0.52938 0.52942

0.8 0.43743 0.43919 0.43916 0.43910 0.43914

1.0 0.37336 0.37446 0.37443 0.37439 0.37442

3.0 0.15015 0.15019 0.15018 0.15018 0.15018

0.75 0.4 0.90111 0.91053 0.91058 0.91031 0.91026

0.6 0.75862 0.76741 0.76739 0.76723 0.76724

0.8 0.64129 0.64750 0.64747 0.64737 0.64740

1.0 0.55187 0.55620 0.55609 0.55603 0.55605

3.0 0.22454 0.22484 0.22483 0.22481 0.22481

We compute

L2

and

L∞

error norms of the LG-RBFs for

v=

1,

M=

10,

∆t=

0.001 at

t=

1, 2

with respectively

c=

0.588, 0.587 and for

v=

0.1,

M=

10,

∆t=

0.01 at

t=

3, 3.5 with respectively

0.6, 0.611. We compare these with the error norms of the backward differentiation formula of order

one (BDF1) reported in [

32

]. The results are tabulated for the both ﬁrst-order schemes (in time) in the

Tables 4and 5

. The tables indicate that the accuracy of LG-RBFs is much better than BDF1 method [

32

].

It can be seen from Tables 4and 5that the error of LG-RBFs gets smaller than the error of BDF1

method when the ﬁnal time increases, as we expect. In Figure 1we show the numerical solutions of

the LG-RBFs which are computed for

ν=

1,

M=

40,

ν=

0.1, 0.01 with

M=

80 and

∆t=

0.0001 at

different ﬁnal times, and at

x=

0

(

1

/

500

)

1. The presence of the steep descent can be seen from the

Figure 1for small values of viscosity. Thus, proposed scheme can exhibit the correct physical behavior

of the problem.

Table 4. Comparison of error norms for different values of ﬁnal times with M=10 and ∆t=0.001.

ν=1t=1t=2

L2L∞L2L∞

BDF-1, M=80 [32] 1.8457 ×10−62.6102 ×10−61.9600 ×10−10 2.7713 ×10−10

LG-RBFs 1.1727 ×10−10 1.5587 ×10−10 4.8120 ×10−14 6.7895×10−14

Table 5. Comparison of error norms for different values of ﬁnal times with M=10 and ∆t=0.01.

ν=0.1 t=3t=3.5

L2L∞L2L∞

BDF-1 [32] 4.2130 ×10−45.9753 ×10−43.0024 ×10−44.2509 ×10−4

LG-RBFs 5.0581 ×10−57.1572 ×10−53.0291 ×10−54.3053 ×10−5

Mathematics 2019,7, 113 9 of 11

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.2

0.4

0.6

0.8

1

1.2

t=0.0

x

u(x,t)

t=0.1

t=0.15

t=0.20

t=0.25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.2

0.4

0.6

0.8

1

1.2

t=0.4

t=0.0

t=0.6

t=0.8

t=1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.2

0.4

0.6

0.8

1

1.2

t=0.4

t=0.6

t=0.0

t=0.8

t=1.0

Figure 1.

Physical behavior of numerical solutions

u(x

,

t)

for

ν=

1,

M=

40 (top left panel) and

ν=

0.1,

M=

80 (top right panel) and

ν=

0.01,

M=

80 (bottom panel) at different ﬁnal times for

∆t=

0.0001.

4. Conclusions

The ﬁrst-order Lie-Group version of Euler method combined with MQ-RBFs method is proposed

to approximate solutions of the one-dimensional Burgers’ equation. The obtained numerical results

are compared with analytical and reported results in the literature for a test example at different

values of viscosity. It is concluded that the numerical solutions of the proposed scheme are effective in

comparison with the analytical and numerical solutions presented in the previous studies. The results

show that the presented scheme can capture the nonlinear steep behavior of the Burgers’ equation in

the case of smaller values of viscosity. The proposed algorithm can also be used for solving other more

general problems.

Funding: This research received no external funding.

Conﬂicts of Interest: The author declares no conﬂict of interest.

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