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 Kuramoto-Sivashinsky 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
- Kuramoto-Sivashinsky equation
- collocation method
- moving mesh partial differential equation
- numerical solution

## 1. Introduction

The Kuramoto-Sivashinsky equation (KSe) is a non-linear fourth order partial differential equation (PDE) discovered separately by Kuramoto and Sivashinsky in the study of non-linear 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 Belousov-Zhabotinsky 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 non-linear 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 B-spline collocation method [9], Lattice Boltzmann method [10], meshless method of lines [11], Fourier spectral method [12] and septic B-spline collocation method [13].

## 2. Grid generation

Generation of an adaptive mesh in the spatial domain is based on the r-refinement 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 arc-length 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 Crank-Nicolson scheme for the KSe is

where

for the linearization of the non-linear 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 Gauss-Legendre 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 non-uniform 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.