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

$$\frac{\partial}{\partial \xi} \left( M \frac{(\partial \dot{x})}{\partial \xi} \right) = -\frac{1}{\tau} \frac{\partial}{\partial \xi} \left( M \frac{\partial x}{\partial \xi} \right) \tag{2}$$

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

<sup>k</sup>¼i�<sup>p</sup> ð Þ Mk

where the parameter p is called the smoothing index which determines the extent of smoothing and is non-negative. γ is non-negative and is called the smoothing index and determines the

þ

un

<sup>x</sup> <sup>þ</sup> ununþ<sup>1</sup> <sup>x</sup> <sup>þ</sup> <sup>u</sup><sup>n</sup>þ<sup>1</sup> xx <sup>þ</sup> unþ<sup>1</sup> xxxx � � <sup>¼</sup> un � <sup>δ</sup><sup>t</sup>

The variable spatial length of each interval is given by Hi where Hi ¼ Xiþ<sup>1</sup>ð Þ� t Xið Þt for

<sup>s</sup> <sup>¼</sup> <sup>x</sup> � Xið Þ<sup>t</sup>

such that sEð Þ 0; 1 for every subinterval of the mesh (11). Define the septic Hermite basis

Consider the mesh on the domain ½ � a; b which is a solution of MMPDE4 given by

i ¼ 1, …, N. For some xE½ � Xið Þt ; Xiþ<sup>1</sup>ð Þt , define the local variable s as

unþ<sup>1</sup> xx <sup>þ</sup> un xx 2 � �

þ

unþ<sup>1</sup> xxxx <sup>þ</sup> <sup>u</sup><sup>n</sup>

<sup>x</sup> <sup>þ</sup> <sup>u</sup>nunþ<sup>1</sup> <sup>x</sup> � ð Þ uux <sup>n</sup> (9)

<sup>2</sup> un

a ¼ X1ð Þt < X2ð Þt < … < XNþ<sup>1</sup>ðÞ¼ t b (11)

. Expression (9) is substituted into (1) and

xx <sup>þ</sup> un

Hið Þ<sup>t</sup> (12)

xxxx � � (10)

2 � �

xxxx

http://dx.doi.org/10.5772/intechopen.71875

265

¼ 0 (8)

P<sup>i</sup>þ<sup>p</sup> k¼i�p

P<sup>i</sup>þ<sup>p</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Numerical Simulation of Wave (Shock Profile) Propagation of the Kuramoto-Sivashinsky Equation Using an…

2 γ 1þγ � �j j <sup>k</sup>�<sup>i</sup>

vuuuut (7)

γ 1�γ � �j j <sup>k</sup>�<sup>i</sup>

monitor function M~ at the grid points are given by

rigidity of the grid.

3. Discretization in time

unþ<sup>1</sup> � <sup>u</sup><sup>n</sup> δt � �

the terms are rearranged to give

unþ<sup>1</sup> <sup>þ</sup>

functions with the local variables s as

The Crank-Nicolson scheme for the KSe is

Me ¼

<sup>þ</sup> ð Þ uux <sup>n</sup>þ<sup>1</sup> <sup>þ</sup> ð Þ uux <sup>n</sup> 2 " #

uux

for the linearization of the non-linear term ð Þ uux <sup>n</sup>þ<sup>1</sup>

δt <sup>2</sup> unþ<sup>1</sup> un

4. Septic Hermite collocation method

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

<sup>n</sup>þ<sup>1</sup> <sup>¼</sup> unþ<sup>1</sup>

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

$$\frac{M\_{i+1} + M\_i}{2\left(\frac{1}{N}\right)^2} (\dot{\mathbf{x}}\_{i+1} - \dot{\mathbf{x}}\_i) - \frac{M\_i + M\_{i-1}}{2\left(\frac{1}{N}\right)^2} (\dot{\mathbf{x}}\_i - \dot{\mathbf{x}}\_{i-1}) = -\frac{E\_i}{\tau},\tag{3}$$

where

$$E\_i = \frac{M\_{i+1} + M\_i}{2\left(\frac{1}{N}\right)^2} (\mathbf{x}\_{i+1} - \mathbf{x}\_i) - \frac{M\_i + M\_{i-1}}{2\left(\frac{1}{N}\right)^2} (\mathbf{x}\_i - \mathbf{x}\_{i-1})\_i \qquad \text{i = 2, \dots, N} \tag{4}$$

$$\mathbf{x}\_1 = a \tag{5} \tag{5}$$

$$\mathbf{x}\_{N+1} = b. \tag{6}$$

The modified monitor function given by

$$M(\mathbf{x}, t) = \left(1 + \alpha^2 \left(\frac{\partial u}{\partial \mathbf{x}}\right)^2 + \alpha^2 \left(\frac{\partial^2 u}{\partial \mathbf{x}^2}\right)^2\right)^{\frac{1}{2}} \tag{6}$$

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 M~ at the grid points are given by

$$\tilde{M} = \sqrt{\frac{\sum\_{k=i-p}^{i+p} (M\_k)^2 \left(\frac{\mathcal{V}}{1+\mathcal{V}}\right)^{|k-i|}}{\sum\_{k=i-p}^{i+p} \left(\frac{\mathcal{V}}{1-\mathcal{V}}\right)^{|k-i|}}} \tag{7}$$

where the parameter p is called the smoothing index which determines the extent of smoothing and is non-negative. γ is non-negative and is called the smoothing index and determines the rigidity of the grid.
