Numerical Simulation of Wave (Shock Profile) Propagation of the KuramotoSivashinsky Equation Using an Adaptive Mesh Method
researcharticle
Denson Muzadziwa^{1}, Stephen T. Sikwila^{2} and Stanford Shateyi^{3}^{∗}
^{[1]} University of Zimbabwe, Harare, Zimbabwe
^{[2]} Sol Plaatje University, Kimberley, South Africa
^{[3]} University of Venda, Thohoyandou, South Africa
^{∗}Corresponding author(s) email:
stanford.shateyi@univen.ac.za
DOI: 10.5772/intechopen.71875
Abstract
In this paper, the KuramotoSivashinsky equation is solved using Hermite collocation method on an adaptive mesh. The method uses seventh order Hermite basis functions on a mesh that is adaptive in space. Numerical experiments are carried out to validate effectiveness of the method.
Keywords: adaptive mesh method, KuramotoSivashinsky equation, collocation method, moving mesh partial differential equation, numerical solution
1. Introduction
The KuramotoSivashinsky equation (KSe) is a nonlinear fourth order partial differential equation (PDE) discovered separately by Kuramoto and Sivashinsky in the study of nonlinear stability of travelling waves. Sivashinsky [1] came up with the equation while modelling small thermal diffusive instabilities in laminar flame fronts. Kuramoto [2, 3, 4, 5] derived the equation in the study of the BelousovZhabotinsky reaction as a model of diffusion induced chaos. The KSe is of interest to many researchers because of its ability to describe several physical contexts such as long waves on thin films or on the interface between two viscous fluids [6] and unstable drift waves in plasmas. The equation is also used as a model to describe spatially uniform oscillating chemical reaction in a homogeneous medium and fluctuations in fluid films on inclines [7]. In one dimension, consider the KSe of the form
The second derivative term is an energy source and thus has a distributing effect. The nonlinear term is a correction to the phase speed and responsible for transferring energy. The fourth derivative term is the dominating term and is responsible for stabilising the equation. Several methods have been used to solve the KSe numerically and these include Chebyshev spectral collocation method [8], Quintic Bspline collocation method [9], Lattice Boltzmann method [10], meshless method of lines [11], Fourier spectral method [12] and septic Bspline collocation method [13].
2. Grid generation
Generation of an adaptive mesh in the spatial domain is based on the rrefinement technique [14] which relocates a fixed number of nodal points to regions which need high spatial resolution in order to capture important characteristics in the solution. This has the benefit of improving computational effort in those regions of interest whilst using a fixed number of mesh points. The relocation of the fixed number of nodal points at any given time is achieved by solving Moving Mesh Partial Differential Equations (MMPDEs) [15, 16] derived from the Equidistribution Principle (EP). The EP [17] makes use of a measure of the solution error called a monitor function, denoted by M which is a positive definite and user defined function of the solution and/or its derivatives. Mesh points are then chosen by equally distributing the error in each subinterval. In this paper, MMPDE4 [15] is chosen to generate the adaptive mesh because of its ability to stabilise mesh trajectories and ability to give unique solutions for the mesh velocities with Dirichlet boundary conditions. MMPDE4 is given by
where
τ
is the relaxation parameter and it plays the role of driving the mesh towards equidistribution. Central finite difference approximation of MMPDE4 in space on the interval
a≤x≤b
gives
where
The modified monitor function given by
is used. It is composed of the standard arclength monitor and the curvature monitor functions. Smoothing on the monitor function is done as described in [15]. Values of the smoothed monitor function
M˜
at the grid points are given by
where the parameter
p
is called the smoothing index which determines the extent of smoothing and is nonnegative.
γ
is nonnegative and is called the smoothing index and determines the rigidity of the grid.
3. Discretization in time
The CrankNicolson scheme for the KSe is
where
δt
is the time step. Rubin and Graves [18] suggested the expression
for the linearization of the nonlinear term
uuxn+1
. Expression (9) is substituted into (1) and the terms are rearranged to give
4. Septic Hermite collocation method
Consider the mesh on the domain
ab
which is a solution of MMPDE4 given by
The variable spatial length of each interval is given by
Hi
where
Hi=Xi+1t−Xit
for
i=1,…,N
. For some
xϵXitXi+1t
, define the local variable
s
as
such that
sϵ01
for every subinterval of the mesh (11). Define the septic Hermite basis functions with the local variables
s
as
For
l=0,1,2,3
the functions
L0,ls
and
L1,ls
yield the following conditions
where
δk,l
denotes the Kronecker delta. The physical solution
uxt
on the mesh (11) is approximated by the piecewise Hermite polynomial [19]
Where
Uit,Ux,it,Uxx,it
and
Uxxx,it
are the unknown variables. Derivatives of
Uxt
with respect to the spatial variable
x
for
x∈XitXi+1t
are obtained by direct differentiation of (14) to give
for
l=1,2,3,4.
In each subinterval
XitXi+1t
of the mesh (11), define four GaussLegendre points
which are given by
One regards these points as the collocation points in each subinterval of the mesh (11). Scaling of the GaussLegendre points into subsequent intervals is done by defining the collocation points as
and redefining the local variable
s
as
for
i=1,…,N
and
j=1,2,3,4
. Evaluation of the Hermite polynomial approximation (14), its first, second and fourth derivatives (15) is then done at the four internal collocation points in each subinterval
XiXi+1
and substitution of the expressions into (10) gives the difference equation
where
and
From the boundary conditions (28) and (29), one gets
which results in a consistent system of
4N+4
equations in
4N+4
unknowns.
5. Solution approach for the PDE system
The PDE system is solved using the rezoning approach which works best with the decoupled solution procedure [20]. The rezoning approach allow varying criteria of convergence for the mesh and physical equation since in practice the mesh does not require the same level of accuracy to compute as compared to the physical solution. The algorithm for the rezoning approach is as follows:
Solve the given physical PDE on the current mesh.
Use the PDE solution obtained to calculate the monitor function.
Find the new mesh by solving a MMPDE.
Adjust the current PDE solution to suite the new mesh by interpolation.
Solve the physical PDE on the new mesh for the solution in the next time.
6. Solution adjustment by interpolation
Discretization of the time domain
tatb
is done using the following finite sequence
At each time
t=tn=n×dt
, consider a nonuniform spatial mesh
Xini=1N+1
given by
where
Xin=Xitn
with
Hin=Xi+1n−Xin
being a nonuniform spatial step for
i=1,…,N
. At the same time step
t=tn
one also considers the approximations to the exact solution
uxt
and its derivatives given by
Uini=1N+1
and
Uilni=1N+1
respectively where
Uiln
represents the
lth
derivative approximation with respect to the variable
x
at the time
t=tn
For
l=1,2,3
. A new mesh
X˜ini=1N+1
is generated by (2) at each current time step
tn
. The goal is to determine the new approximations
U˜ini=1N+1
and
U˜ilni=1N+1
which are related to the new mesh
X˜ini=1N+1
in a similar manner the approximations
Uini=1N+1
and
Uilni=1N+1
are related to the old mesh
Xini=1N+1
in each subinterval
XiXi+1
. This process of updating the solution and its derivatives from the old mesh to the new mesh is achieved by interpolation. One considers the septic Hermite interpolating polynomial, a piecewise polynomial which allows the function values and its three consecutive derivatives to be satisfied in each subinterval
XiXi+1
. The Hermite polynomial (14) is written in compact form as
where the
4N+1
unknowns are given by
Given the partition (23) and approximations
Uiln
for
l=0,1,2,3
, suppose interpolation of
Ulx
is required at
x=Xi˜n
where
Xi˜n∈XinXi+1n
for
i=1,…,N
. Firstly, the local coordinate
s
of
Xi˜n
is defined as
U˜lX˜in
is then defined as
for
l=0,1,2,3
to give the interpolated values of
U˜
and the first three consecutive derivatives on the new subinterval
X˜inX˜i+1n
. In order to compute the approximations of
U
at the next time step
t=tn+1
denoted by
Uini=1N+1
, the values of the new mesh
X˜ini=1N+1
and the updated approximations
U˜ini=1N+1
are used in a septic Hermite collocation numerical scheme. The new approximations
Uin+1i=1N+1
and the new mesh
X˜in+1i=1N+1
become the starting conditions for repeating the whole adaptive process.
7. Numerical results
Consider the KSe
in the domain
−3030,t>0
with boundary conditions
Where
σ,β,ω
and
ζ
are obtained from the exact solution
With
c=0.1,x0=−12
and
k=121119
.
Figures 1 and 2 show the behaviour of the numerical solution and the absolute error, respectively of the KSe equation on a stationary mesh using Hermite collocation method at
t=4
with
N=100
and
δt=0.001
. In Figure 1, one observes that the numerical solution tracks the exact solution with the absolute error variation as shown in Figure 2.
Figure 1.
Hermite collocation method, uniform mesh, numerical solution behaviour of KSe at
t=4
with
N=100
and
δt=0.001
.
Figure 2.
Hermite collocation method, uniform mesh, absolute error in numerical solution of KSe at
t=4,N=100
and
δt=0.001
.
Figure 3 shows the solution obtained by the collocation method on a stationary mesh for time
t=0,1,2,3,4
. The movement of the solution is from left to right as time increases and the solution tracks the exact solution with no oscillations. One also observes that the concentration of mesh points is higher in the flatter regions of the solution profile in comparison to the concentration in the steeper region.
Figure 3.
Hermite collocation method, stationary mesh, numerical solution behaviour of KSe problem with
N=100,δt=0.001
up to final time
T=4.
Figures 4 and 5 show the numerical solution profile and the behaviour of the maximum absolute error, respectively at
t=4
with
N=100,δt=0.001
and
α=8
on an adaptive mesh. In Figure 4, one observes that the numerical solution is able to track the exact solution and the distribution of mesh points is almost equal along the solution profile which enables resolution of the solution with minimum errors.
Figure 4.
Hermite collocation method, nonuniform mesh, numerical solution behaviour of KSe problem at
t=4
with
N=100,δt=0.001,τ=2×10−2
and
α=8
.
Figure 5.
Hermite collocation method, nonuniform mesh, absolute error in numerical solution of KSe at
t=100,δt=0.001,τ=2×10−2
and
α=8
.
Figure 6 shows the numerical solution profiles produced by the adaptive collocation method for time
t=0,1,2,3,4
. One observes that the solution moves from left to right as time progresses. The mesh points at different times keep on tracking the solution profile and maintain an almost equal distribution along the profile up to final time
T=4
. Figure 7 shows the paths taken by the mesh points in tracking the solution profile. In Table 1, the infinity norm error for an adaptive collocation method is calculated and results are compared with the method in [13]. Results show improvements in the maximum point wise errors when an adaptive Hermite collocation method is used.
Figure 6.
Hermite collocation method, adaptive mesh, numerical solution behaviour of KSe up to final time
T=4
for
N=100,δt=0.001,τ=2×10−2
and
α=8
.
Figure 7.
Hermite collocation method, mesh trajectories of KSe equation up to final time
T=4
with
N=100,δt=0.001,τ=2×10−2
and
α=8
.
Time  Hermite collocation  Method in [19] 

0.5 
9.0×10−4

1.03619×10−3

1 
1.4×10−3

1.63762×10−3

1.5 
1.9×10−3

2.07273×10−3

2 
1.7×10−3

2.48375×10−3

2.5 
2.0×10−3

2.79434×10−3

3 
2.1×10−3

3.00439×10−3

3.5 
2.1×10−3

3.16038×10−3

4 
2.1×10−3

3.43704×10−3

Table 1.
Comparison of maximum pointwise errors in the numerical solution of the KSe on an adaptive mesh at different times with
δt=0.001
and
N=100
.
8. Conclusions
The KSe is solved using an adaptive mesh method with discretization in the spatial domain done using seventh order Hermite basis functions. Numerical results show that Hermite collocation method on a nonuniform adaptive mesh is able to improve the accuracy of the numerical solution of the KSe. The method is able to keep track of the region of rapid solution variation in the KSe, which is one of the desired properties of an adaptive mesh method.