3. Discretization in time

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 B-spline collocation method [9], Lattice Boltzmann method [10], meshless method of lines [11], Fourier spectral method [12] and septic B-spline

264 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

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

> ¼ � <sup>1</sup> τ ∂ <sup>∂</sup><sup>ξ</sup> <sup>M</sup> <sup>∂</sup><sup>x</sup> ∂ξ � �

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

> Mi þ Mi�<sup>1</sup> 2 <sup>1</sup> N

Mi þ Mi�<sup>1</sup> 2 <sup>1</sup> N

> ∂x � �<sup>2</sup>

� �<sup>2</sup> !<sup>1</sup>

� �<sup>2</sup> <sup>x</sup>\_<sup>i</sup> � <sup>x</sup>\_ ð Þ¼� <sup>i</sup>�<sup>1</sup>

x<sup>1</sup> ¼ a xNþ<sup>1</sup> ¼ b: (5)

2

<sup>þ</sup> <sup>α</sup><sup>2</sup> <sup>∂</sup><sup>2</sup><sup>u</sup> ∂x<sup>2</sup> Ei

� �<sup>2</sup> ð Þ xi � xi�<sup>1</sup> , i <sup>¼</sup> <sup>2</sup>, …, N (4)

<sup>τ</sup> , (3)

(2)

(6)

velocities with Dirichlet boundary conditions. MMPDE4 is given by ∂ <sup>∂</sup><sup>ξ</sup> <sup>M</sup> ð Þ <sup>∂</sup>x\_ ∂ξ � �

� �<sup>2</sup> <sup>x</sup>\_<sup>i</sup>þ<sup>1</sup> � <sup>x</sup>\_ ð Þ�<sup>i</sup>

� �<sup>2</sup> ð Þ� xiþ<sup>1</sup> � xi

M xð Þ¼ ; <sup>t</sup> <sup>1</sup> <sup>þ</sup> <sup>α</sup><sup>2</sup> <sup>∂</sup><sup>u</sup>

Miþ<sup>1</sup> þ Mi 2 <sup>1</sup> N

Ei <sup>¼</sup> Miþ<sup>1</sup> <sup>þ</sup> Mi 2 <sup>1</sup> N

The modified monitor function given by

collocation method [13].

2. Grid generation

a ≤ x ≤ b gives

where

The Crank-Nicolson scheme for the KSe is

$$
\left[\frac{\mathbf{u}^{n+1} - \mathbf{u}^{n}}{\delta t}\right] + \left[\frac{\left(\mathbf{u}\mathbf{u}\_{x}\right)^{n+1} + \left(\mathbf{u}\mathbf{u}\_{x}\right)^{n}}{2}\right] + \left[\frac{\mathbf{u}\_{xx}^{n+1} + \mathbf{u}\_{xx}^{n}}{2}\right] + \left[\frac{\mathbf{u}\_{xxxx}^{n+1} + \mathbf{u}\_{xxxx}^{n}}{2}\right] = \mathbf{0} \tag{8}
$$

where δt is the time step. Rubin and Graves [18] suggested the expression

$$
\mu u\_{\mathbf{x}}{}^{n+1} = \mu^{n+1} u\_{\mathbf{x}}^{n} + \mu^{n} u\_{\mathbf{x}}^{n+1} - (\mu u\_{\mathbf{x}})^{n} \tag{9}
$$

for the linearization of the non-linear term ð Þ uux <sup>n</sup>þ<sup>1</sup> . Expression (9) is substituted into (1) and the terms are rearranged to give

$$u^{n+1} + \frac{\delta t}{2} \left[ u^{n+1} u\_x^n + u^n u\_x^{n+1} + u\_{xx}^{n+1} + u\_{xxx}^{n+1} \right] = u^n - \frac{\delta t}{2} \left[ u\_{xx}^n + u\_{xxx}^n \right] \tag{10}$$
