Comparison of maximum pointwise errors in the numerical solution of the KSe on an adaptive mesh at different times with
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
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
where the parameter
3. Discretization in time
The CrankNicolson scheme for the KSe is
where
for the linearization of the nonlinear term
4. Septic Hermite collocation method
Consider the mesh on the domain
The variable spatial length of each interval is given by
such that
For
where
Where
for
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
for
where
and
From the boundary conditions (28) and (29), one gets
which results in a consistent system of
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
At each time
where
where the
Given the partition (23) and approximations
for
7. Numerical results
Consider the KSe
in the domain
Where
With
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
Figure 3 shows the solution obtained by the collocation method on a stationary mesh for time
Figures 4 and 5 show the numerical solution profile and the behaviour of the maximum absolute error, respectively at
Figure 6 shows the numerical solution profiles produced by the adaptive collocation method for time
Time  Hermite collocation  Method in [19] 

0.5 


1 


1.5 


2 


2.5 


3 


3.5 


4 


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.
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