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A Meshless Method for Burgers’ Equation Using Multiquadric Radial Basis Functions With a Lie-Group Integrator

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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.
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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) = v2u(x,t)
x2(1)
with initial condition
u(x, 0)=u0(x),axb
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 [711].
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=
xxj
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(xixj)2+c2,
where the collocation points
{xi}
on the interval [a, b] are called for 2
iM
2 as center points and
for
i=
1 and
i=M
as boundary points. If one inserts Equation
(3)
for 2
iM
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=M1
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=M1
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.ukuk2=kuk2kuk2=0, (12)
where Minkowski metric gis defined as
g="Id×d0d×1
01×d1#,
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
UnU(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+1SO0(d, 1). On the other hand, it is easy to express exp(tA(tn)) in closed-form
exp(tA(tn)) =
In+(an1)
kfnk2fnfT
n
bnfn
kfnk
bnfT
n
kfnkan
,
where , unu(tn),t=tn+1tn,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=(an1)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=nt,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, . . . , M1,
where
f0=hu00
0(xi)F0(u0(xi))iM1
i=2
and
ζ0
can be computed from Equation (15). For 1
nN
1,
i=2, . . . , M1 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=M1
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, . . . , M1 as
b1
i=u0(xi) + ζ0f0,
bn+1
i=
M
j=1
λn
jϕrij +λn
M+1xi+λn
M+2+ζnfn,n=1, . . . , N1,
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
D1
.
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[1cos(πx)]odx,
ki=2Z1
0exp n(2πν)1[1cos(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 Lnorm.
uapr u
L2= h
M
j=1
(uj,Nu(xj,t))2!1/2
,
uapr u
L=max
1jMuj,Nu(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
L2LL2L
BDF-1, M=80 [32] 1.8457 ×1062.6102 ×1061.9600 ×1010 2.7713 ×1010
LG-RBFs 1.1727 ×1010 1.5587 ×1010 4.8120 ×1014 6.7895×1014
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
L2LL2L
BDF-1 [32] 4.2130 ×1045.9753 ×1043.0024 ×1044.2509 ×104
LG-RBFs 5.0581 ×1057.1572 ×1053.0291 ×1054.3053 ×105
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.
References
1.
Bateman, H. Some recent researches on the motion of fluids. Mon. Weather Rev.
1915
,43, 163–170,
doi:10.1175/1520-0493(1915)43<163:SRROTM>2.0.CO;2.
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... It describes the phenomena such as dispersion in porous media, weak shock propagation, heat conduction, acoustic attenuation in fog, compressible turbulence, gas-dynamics, continuous stochastic processes, and even continuum traffic simulation. Burgers' equation is as follows [1][2][3][4][5][6]: ...
... In view of the universality of Burgers' equation in describing lots of important physical phenomena, many numerical methods were introduced to solve it such as the finite difference method, finite element method, mixed finite element method, characteristics mixed finite element, spectral method, and meshless method; see [1][2][3][4][5][6] and the references therein. ...
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