4. Septic Hermite collocation method

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

$$a = X\_1(t) < X\_2(t) < \dots < X\_{N+1}(t) = b \tag{11}$$

The variable spatial length of each interval is given by Hi where Hi ¼ Xiþ<sup>1</sup>ð Þ� t Xið Þt for i ¼ 1, …, N. For some xE½ � Xið Þt ; Xiþ<sup>1</sup>ð Þt , define the local variable s as

$$s = \frac{x - X\_i(t)}{H\_i(t)}\tag{12}$$

such that sEð Þ 0; 1 for every subinterval of the mesh (11). Define the septic Hermite basis functions with the local variables s as

$$\begin{aligned} L\_{0,0} &= \left(20s^3 + 10s^2 + 4s + 1\right)(s - 1)^4 \\ L\_{0,1} &= s\left(10s^2 + 4s + 1\right)(s - 1)^4 \\ L\_{0,2} &= \frac{s^2}{2}(4s + 1)(s - 1)^4 \\ L\_{0,3} &= \frac{s^3}{6}(s - 1)^4 \\ L\_{1,0} &= -\left(20s^3 - 70s^2 + 84s - 35\right)s^4 \\ L\_{1,2} &= -\frac{s^4}{2}(s - 1)^2(4s - 5) \\ L\_{1,3} &= \frac{s^4}{6}(s - 1)^3 \end{aligned} \tag{13}$$

<sup>r</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup> 2 �

<sup>r</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup> 2 �

> s ð Þi

� � <sup>þ</sup> <sup>U</sup><sup>n</sup>

00 <sup>0</sup>, <sup>1</sup> sj � � <sup>þ</sup> <sup>U</sup><sup>n</sup>

<sup>0</sup>,<sup>1</sup> sj � � <sup>þ</sup> <sup>U</sup><sup>n</sup>

<sup>1</sup>, <sup>1</sup> sj � � <sup>þ</sup> <sup>U</sup><sup>n</sup>

xx,i ð Þ<sup>t</sup> <sup>H</sup><sup>2</sup>

� � <sup>þ</sup> <sup>U</sup><sup>n</sup>

<sup>i</sup> ð Þt L0,<sup>2</sup> sj

xx,iþ<sup>1</sup>Hið Þ<sup>t</sup> <sup>L</sup>1, <sup>1</sup> sj

00 <sup>1</sup>, <sup>1</sup> sj � � <sup>þ</sup> <sup>U</sup><sup>n</sup>

<sup>i</sup> ð Þ<sup>t</sup> <sup>L</sup>ð Þ iv <sup>0</sup>,<sup>2</sup> sj � � <sup>þ</sup> <sup>U</sup><sup>n</sup>

<sup>1</sup>, <sup>1</sup> sj � � <sup>þ</sup> <sup>U</sup><sup>n</sup>

xx,i ð Þ<sup>t</sup> <sup>H</sup><sup>2</sup> <sup>i</sup> ð Þt L 00 <sup>0</sup>, <sup>2</sup> sj � � <sup>þ</sup> <sup>U</sup><sup>n</sup>

� � h

xx,iþ<sup>1</sup>Hið Þ<sup>t</sup> <sup>L</sup>

xx,i ð Þ<sup>t</sup> <sup>H</sup><sup>2</sup>

xx,iþ<sup>1</sup>Hið Þ<sup>t</sup> <sup>L</sup>ð Þ iv

� � h

points as

equation

β ð Þi <sup>j</sup><sup>1</sup> <sup>U</sup><sup>n</sup>þ<sup>1</sup> <sup>i</sup> þ β ð Þi <sup>j</sup><sup>2</sup> <sup>U</sup><sup>n</sup>þ<sup>1</sup> x,i þ β ð Þi <sup>j</sup><sup>3</sup> <sup>U</sup><sup>n</sup>þ<sup>1</sup> xx,i þ β ð Þi <sup>j</sup><sup>4</sup> <sup>U</sup><sup>n</sup>þ<sup>1</sup> xxx,i þ β ð Þi <sup>j</sup><sup>5</sup> <sup>U</sup><sup>n</sup>þ<sup>1</sup> <sup>i</sup>þ<sup>1</sup> <sup>þ</sup> <sup>β</sup> ð Þi <sup>j</sup><sup>6</sup> <sup>U</sup><sup>n</sup>þ<sup>1</sup> x,iþ<sup>1</sup> <sup>þ</sup> <sup>β</sup>

where

and

ψn ij <sup>¼</sup> <sup>U</sup><sup>n</sup>

<sup>i</sup> ð Þt L0, <sup>0</sup> sj

<sup>i</sup>þ<sup>1</sup>ð Þ<sup>t</sup> <sup>L</sup>1, <sup>0</sup> sj

þU<sup>n</sup>

� <sup>δ</sup><sup>t</sup> 2H<sup>2</sup> i Un <sup>i</sup> ð Þt L 00 <sup>0</sup>, <sup>0</sup> sj � � <sup>þ</sup> <sup>U</sup><sup>n</sup>

þU<sup>n</sup> <sup>i</sup>þ<sup>1</sup>ð Þ<sup>t</sup> <sup>L</sup> 00 <sup>1</sup>, <sup>0</sup> sj � � <sup>þ</sup> <sup>U</sup><sup>n</sup>

� <sup>δ</sup><sup>t</sup> 2H<sup>4</sup> i Un <sup>i</sup> ð Þ<sup>t</sup> <sup>L</sup>ð Þ iv <sup>0</sup>, <sup>0</sup> sj � � <sup>þ</sup> <sup>U</sup><sup>n</sup>

þU<sup>n</sup>

<sup>i</sup>þ<sup>1</sup>ð Þ<sup>t</sup> <sup>L</sup>ð Þ iv <sup>1</sup>, <sup>0</sup> sj � � <sup>þ</sup> <sup>U</sup><sup>n</sup>

� � <sup>þ</sup> <sup>U</sup><sup>n</sup>

� � <sup>þ</sup> <sup>U</sup><sup>n</sup>

x,i Hið Þt L0, <sup>1</sup> sj

x,iþ<sup>1</sup>Hið Þ<sup>t</sup> <sup>L</sup>1, <sup>1</sup> sj

x,i Hið Þt L

00 <sup>1</sup>,<sup>1</sup> sj � � <sup>þ</sup> <sup>U</sup><sup>n</sup>

x,i Hið Þ<sup>t</sup> <sup>L</sup>ð Þ iv

x,iþ<sup>1</sup>Hið Þ<sup>t</sup> <sup>L</sup>

x,iþ<sup>1</sup>Hið Þ<sup>t</sup> <sup>L</sup>ð Þ iv

and redefining the local variable s as

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>525</sup> <sup>þ</sup> <sup>70</sup> ffiffiffiffiffi <sup>30</sup> <sup>p</sup> <sup>p</sup> 70

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

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>525</sup> � <sup>70</sup> ffiffiffiffiffi <sup>30</sup> <sup>p</sup> <sup>p</sup> 70

Xij ¼ Xi þ Hirj, i ¼ 1, …:, N, j ¼ 1, 2, 3, 4: (16)

ð Þi <sup>j</sup><sup>7</sup> <sup>U</sup><sup>n</sup>þ<sup>1</sup> xx,iþ<sup>1</sup> <sup>þ</sup> <sup>β</sup>

� � <sup>þ</sup> <sup>U</sup><sup>n</sup>

xxx,i ð Þ<sup>t</sup> <sup>H</sup><sup>3</sup>

� � <sup>þ</sup> <sup>U</sup><sup>n</sup>

ð Þi <sup>j</sup><sup>8</sup> <sup>U</sup><sup>n</sup>þ<sup>1</sup>

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

<sup>i</sup> ð Þt L0, <sup>3</sup> sj � �

xxx,iþ<sup>1</sup>Hið Þ<sup>t</sup> <sup>L</sup>1, <sup>1</sup> sj

00 <sup>1</sup>, <sup>1</sup> sj � ��

> <sup>1</sup>,<sup>1</sup> sj � ��

> > (19)

xxx,i ð Þ<sup>t</sup> <sup>H</sup><sup>3</sup> <sup>i</sup> ð Þt L 00 <sup>0</sup>, <sup>3</sup> sj

xxx,iþ<sup>1</sup>Hið Þ<sup>t</sup> <sup>L</sup>

xxx,i ð Þ<sup>t</sup> <sup>H</sup><sup>3</sup> <sup>i</sup> ð Þ<sup>t</sup> Liv <sup>0</sup>,<sup>3</sup> sj

xxx,iþ<sup>1</sup>Hið Þ<sup>t</sup> <sup>L</sup>ð Þ iv

xxx,iþ<sup>1</sup> <sup>¼</sup> <sup>ψ</sup><sup>n</sup>

� �

ij (18)

(17)

267

r<sup>3</sup> ¼ 1 � r<sup>1</sup>

r<sup>4</sup> ¼ 1 � r<sup>2</sup>

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

> <sup>j</sup> <sup>¼</sup> Xij � Xi Hi

for i ¼ 1, …, N and j ¼ 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 Xi ½ � ; Xiþ<sup>1</sup> and substitution of the expressions into (10) gives the difference

For l ¼ 0, 1, 2, 3 the functions L0,lð Þs and L1,lð Þs yield the following conditions

$$\frac{d^k}{ds^k}L\_{0,l}(0) = \delta\_{k,l\prime} \qquad \qquad \frac{d^k}{ds^k}L\_{0,l}(1) = 0, \qquad \qquad k,l = 0,1,2,3$$

$$\frac{d^k}{ds^k}L\_{0,l}(0) = 0, \qquad \qquad \frac{d^k}{ds^k}L\_{1,l}(1) = \delta\_{k,l\prime} \qquad \qquad k,l = 0,1,2,3$$

where δk,l denotes the Kronecker delta. The physical solution u xð Þ ; t on the mesh (11) is approximated by the piecewise Hermite polynomial [19]

$$\begin{aligned} \mathcal{U}(\mathbf{x},t) &= \mathcal{U}\_{\mathrm{i}}(t)\mathcal{L}\_{\mathrm{0},\mathrm{0}}(\mathbf{s}) + \mathcal{U}\_{\mathrm{x},i}H\_{\mathrm{i}}(t)\mathcal{L}\_{\mathrm{0},\mathrm{1}}(\mathbf{s}) + \mathcal{U}\_{\mathrm{xx},\mathrm{i}}(t)H\_{\mathrm{i}}^{2}(t)\mathcal{L}\_{\mathrm{0},\mathrm{2}}(\mathbf{s}) + \mathcal{U}\_{\mathrm{xxx},\mathrm{i}}(t)H\_{\mathrm{i}}^{3}(t)\mathcal{L}\_{\mathrm{0},\mathrm{3}}(\mathbf{s}) \\ &+ \mathcal{U}\_{\mathrm{i}+1}(t)\mathcal{L}\_{\mathrm{1},\mathrm{0}}(\mathbf{s}) + \mathcal{U}\_{\mathrm{x},i+1}H\_{\mathrm{i}}(t)\mathcal{L}\_{\mathrm{1},\mathrm{1}}(\mathbf{s}) + \mathcal{U}\_{\mathrm{xx},i+1}(t)H\_{\mathrm{i}}^{2}(t)\mathcal{L}\_{\mathrm{1},\mathrm{2}}(\mathbf{s}) + \mathcal{U}\_{\mathrm{xxx},i+1}(t)H\_{\mathrm{i}}^{3}(t)\mathcal{L}\_{\mathrm{1},\mathrm{3}}(\mathbf{s}), \end{aligned} \tag{14}$$

Where Uið Þt , Ux,ið Þt , Uxx,ið Þt and Uxxx,ið Þt are the unknown variables. Derivatives of U xð Þ ; t with respect to the spatial variable x for x∈ ½ � Xið Þt ; Xiþ<sup>1</sup>ð Þt are obtained by direct differentiation of (14) to give

$$\begin{split} \frac{d^{(l)}L^{(l)}(x,t)}{d\mathbf{x}^{(l)}} &= \frac{1}{H\_i(t)^{(l)}} \Big[ \boldsymbol{U}\_i(t) \frac{d^{(l)}\boldsymbol{L}\_{0,0}}{d\mathbf{s}^{(l)}} + \boldsymbol{U}\_{x,i}(t)\boldsymbol{H}\_i(t) \frac{d^{(l)}\boldsymbol{L}\_{0,1}}{d\mathbf{s}^{(l)}} + \boldsymbol{U}\_{xx,i}(t)\boldsymbol{H}\_i^2(t) \frac{d^{(l)}\boldsymbol{L}\_{0,2}}{d\mathbf{s}^{(l)}} \\ &+ \boldsymbol{U}\_{xx,i}(t)\boldsymbol{H}\_i^3(t) \frac{d^{(l)}\boldsymbol{L}\_{0,3}}{d\mathbf{s}^{(l)}} + \boldsymbol{U}\_{i+1}(t) \frac{d^{(l)}\boldsymbol{L}\_{1,0}}{d\mathbf{s}^{(l)}} + \boldsymbol{U}\_{x,i+1}(t)\boldsymbol{H}\_i(t) \frac{d^{(l)}\boldsymbol{L}\_{1,1}}{d\mathbf{s}^{(l)}} \\ &+ \boldsymbol{U}\_{xx,i+1}(t)\boldsymbol{H}\_i^2(t) \frac{d^{(l)}\boldsymbol{L}\_{1,2}}{d\mathbf{s}^{(l)}} + \boldsymbol{U}\_{xx,i+1}(t)\boldsymbol{H}\_i^3(t) \frac{d^{(l)}\boldsymbol{L}\_{1,3}}{d\mathbf{s}^{(l)}} \Bigg] \end{split} \tag{15}$$

for l ¼ 1, 2, 3, 4: In each subinterval ½ � Xið Þt ; Xiþ<sup>1</sup>ð Þt of the mesh (11), define four Gauss-Legendre points

$$0 < \rho\_1 < \rho\_2 < \rho\_3 < \rho\_4 < 1$$

which are given by

Numerical Simulation of Wave (Shock Profile) Propagation of the Kuramoto-Sivashinsky Equation Using an… http://dx.doi.org/10.5772/intechopen.71875 267

$$\rho\_1 = \frac{1}{2} - \frac{\sqrt{525 + 70\sqrt{30}}}{70}$$

$$\rho\_2 = \frac{1}{2} - \frac{\sqrt{525 - 70\sqrt{30}}}{70}$$

$$\rho\_3 = 1 - \rho\_1$$

$$\rho\_4 = 1 - \rho\_2$$

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

Xij ¼ Xi þ Hirj, i ¼ 1, …:, N, j ¼ 1, 2, 3, 4: (16)

and redefining the local variable s as

$$s\_j^{(i)} = \frac{\mathbf{X}\_{ij} - \mathbf{X}\_i}{H\_i} \tag{17}$$

for i ¼ 1, …, N and j ¼ 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 Xi ½ � ; Xiþ<sup>1</sup> and substitution of the expressions into (10) gives the difference equation

$$\boldsymbol{\beta}\_{\text{j}}^{(i)}\mathbf{U}\_{i}^{n+1} + \boldsymbol{\beta}\_{\text{j}}^{(i)}\mathbf{U}\_{\text{x},i}^{n+1} + \boldsymbol{\beta}\_{\text{j}}^{(i)}\mathbf{U}\_{\text{xx},i}^{n+1} + \boldsymbol{\beta}\_{\text{j}}^{(i)}\mathbf{U}\_{\text{xxx},i}^{n+1} + \boldsymbol{\beta}\_{\text{j}}^{(i)}\mathbf{U}\_{i+1}^{n+1} + \boldsymbol{\beta}\_{\text{j}}^{(i)}\mathbf{U}\_{\text{x},i+1}^{n+1} + \boldsymbol{\beta}\_{\text{j}}^{(i)}\mathbf{U}\_{\text{xxx},i+1}^{n+1} = \boldsymbol{\psi}\_{\text{y}}^{n} \tag{18}$$

where

L0,<sup>0</sup> ¼ 20s

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

L0,<sup>1</sup> ¼ s 10s

<sup>L</sup>0,<sup>2</sup> <sup>¼</sup> <sup>s</sup><sup>2</sup>

<sup>L</sup>0,<sup>3</sup> <sup>¼</sup> <sup>s</sup><sup>3</sup>

L1,<sup>0</sup> ¼ � 20s

<sup>L</sup>1,<sup>2</sup> ¼ � <sup>s</sup><sup>4</sup>

<sup>L</sup>1,<sup>3</sup> <sup>¼</sup> <sup>s</sup><sup>4</sup>

dsk <sup>L</sup>0,lð Þ¼ <sup>0</sup> <sup>δ</sup>k,l, dk

dsk <sup>L</sup>0,lð Þ¼ <sup>0</sup> <sup>0</sup>, dk

<sup>þ</sup>Uiþ<sup>1</sup>ðtÞL1, <sup>0</sup>ðsÞ þ Ux,iþ<sup>1</sup>HiðtÞL1,1ðsÞ þ Uxx,iþ<sup>1</sup>ðtÞH<sup>2</sup>

<sup>ð</sup>l<sup>Þ</sup> Uiðt<sup>Þ</sup>

"

dðl<sup>Þ</sup> L0,<sup>0</sup>

<sup>i</sup> ðtÞ dðl<sup>Þ</sup> L0,<sup>3</sup> dsðl<sup>Þ</sup> <sup>þ</sup> Uiþ<sup>1</sup>ðt<sup>Þ</sup>

<sup>i</sup> ðtÞ dðl<sup>Þ</sup> L1, <sup>2</sup>

approximated by the piecewise Hermite polynomial [19]

<sup>U</sup>ðx;tÞ ¼ UiðtÞL0,0ðsÞ þ Ux,iHiðtÞL0, <sup>1</sup>ðsÞ þ Uxx,iðtÞH<sup>2</sup>

HiðtÞ

<sup>þ</sup> Uxxx,iðtÞH<sup>3</sup>

<sup>þ</sup> Uxx,iþ<sup>1</sup>ðtÞH<sup>2</sup>

dk

dk

of (14) to give

points

which are given by

∂ðl<sup>Þ</sup> Uðx;tÞ <sup>∂</sup>xðl<sup>Þ</sup> <sup>¼</sup> <sup>1</sup> <sup>3</sup> <sup>þ</sup> <sup>10</sup><sup>s</sup>

<sup>2</sup> ð Þ <sup>4</sup><sup>s</sup> <sup>þ</sup> <sup>1</sup> ð Þ <sup>s</sup> � <sup>1</sup> <sup>4</sup>

<sup>3</sup> � <sup>70</sup><sup>s</sup>

where δk,l denotes the Kronecker delta. The physical solution u xð Þ ; t on the mesh (11) is

Where Uið Þt , Ux,ið Þt , Uxx,ið Þt and Uxxx,ið Þt are the unknown variables. Derivatives of U xð Þ ; t with respect to the spatial variable x for x∈ ½ � Xið Þt ; Xiþ<sup>1</sup>ð Þt are obtained by direct differentiation

dsðl<sup>Þ</sup> <sup>þ</sup> Ux,iðtÞHiðt<sup>Þ</sup>

dsðl<sup>Þ</sup> <sup>þ</sup> Uxxx,iþ<sup>1</sup>ðtÞH<sup>3</sup>

for l ¼ 1, 2, 3, 4: In each subinterval ½ � Xið Þt ; Xiþ<sup>1</sup>ð Þt of the mesh (11), define four Gauss-Legendre

0 < r<sup>1</sup> < r<sup>2</sup> < r<sup>3</sup> < r<sup>4</sup> < 1

<sup>2</sup> ð Þ <sup>s</sup> � <sup>1</sup> <sup>2</sup>

<sup>6</sup> ð Þ <sup>s</sup> � <sup>1</sup> <sup>3</sup>

For l ¼ 0, 1, 2, 3 the functions L0,lð Þs and L1,lð Þs yield the following conditions

<sup>2</sup> <sup>þ</sup> <sup>84</sup><sup>s</sup> � <sup>35</sup> � �<sup>s</sup>

ð Þ 4s � 5

4

dsk <sup>L</sup>0,lð Þ¼ <sup>1</sup> <sup>0</sup>, k, l <sup>¼</sup> <sup>0</sup>, <sup>1</sup>, <sup>2</sup>, <sup>3</sup>

dsk <sup>L</sup>1,lð Þ¼ <sup>1</sup> <sup>δ</sup>k,l, k, l <sup>¼</sup> <sup>0</sup>, <sup>1</sup>, <sup>2</sup>, <sup>3</sup>

<sup>i</sup> <sup>ð</sup>tÞL0, <sup>2</sup>ðsÞ þ Uxxx,iðtÞH<sup>3</sup>

dðl<sup>Þ</sup> L0, <sup>1</sup>

dðl<sup>Þ</sup> L1,<sup>0</sup> <sup>i</sup> ðtÞL0, <sup>3</sup>ðsÞ

<sup>i</sup> ðtÞ dðl<sup>Þ</sup> L0,<sup>2</sup> dsðl<sup>Þ</sup>

> dðl<sup>Þ</sup> L1, <sup>1</sup> dsðl<sup>Þ</sup>

<sup>i</sup> <sup>ð</sup>tÞL1, <sup>3</sup>ðsÞ, (14)

(15)

<sup>i</sup> <sup>ð</sup>tÞL1, <sup>2</sup>ðsÞ þ Uxxx,iþ<sup>1</sup>ðtÞH<sup>3</sup>

dsðl<sup>Þ</sup> <sup>þ</sup> Uxx,iðtÞH<sup>2</sup>

dsðl<sup>Þ</sup> <sup>þ</sup> Ux,iþ<sup>1</sup>ðtÞHiðt<sup>Þ</sup>

#

<sup>i</sup> ðtÞ dðl<sup>Þ</sup> L1,<sup>3</sup> dsðl<sup>Þ</sup>

(13)

<sup>6</sup> ð Þ <sup>s</sup> � <sup>1</sup> <sup>4</sup>

<sup>2</sup> <sup>þ</sup> <sup>4</sup><sup>s</sup> <sup>þ</sup> <sup>1</sup> � �ð Þ <sup>s</sup> � <sup>1</sup> <sup>4</sup>

<sup>2</sup> <sup>þ</sup> <sup>4</sup><sup>s</sup> <sup>þ</sup> <sup>1</sup> � �ð Þ <sup>s</sup> � <sup>1</sup> <sup>4</sup>

ψn ij <sup>¼</sup> <sup>U</sup><sup>n</sup> <sup>i</sup> ð Þt L0, <sup>0</sup> sj � � <sup>þ</sup> <sup>U</sup><sup>n</sup> x,i Hið Þt L0, <sup>1</sup> sj � � <sup>þ</sup> <sup>U</sup><sup>n</sup> xx,i ð Þ<sup>t</sup> <sup>H</sup><sup>2</sup> <sup>i</sup> ð Þt L0,<sup>2</sup> sj � � <sup>þ</sup> <sup>U</sup><sup>n</sup> xxx,i ð Þ<sup>t</sup> <sup>H</sup><sup>3</sup> <sup>i</sup> ð Þt L0, <sup>3</sup> sj � � þU<sup>n</sup> <sup>i</sup>þ<sup>1</sup>ð Þ<sup>t</sup> <sup>L</sup>1, <sup>0</sup> sj � � <sup>þ</sup> <sup>U</sup><sup>n</sup> x,iþ<sup>1</sup>Hið Þ<sup>t</sup> <sup>L</sup>1, <sup>1</sup> sj � � <sup>þ</sup> <sup>U</sup><sup>n</sup> xx,iþ<sup>1</sup>Hið Þ<sup>t</sup> <sup>L</sup>1, <sup>1</sup> sj � � <sup>þ</sup> <sup>U</sup><sup>n</sup> xxx,iþ<sup>1</sup>Hið Þ<sup>t</sup> <sup>L</sup>1, <sup>1</sup> sj � � � <sup>δ</sup><sup>t</sup> 2H<sup>2</sup> i Un <sup>i</sup> ð Þt L 00 <sup>0</sup>, <sup>0</sup> sj � � <sup>þ</sup> <sup>U</sup><sup>n</sup> x,i Hið Þt L 00 <sup>0</sup>, <sup>1</sup> sj � � <sup>þ</sup> <sup>U</sup><sup>n</sup> xx,i ð Þ<sup>t</sup> <sup>H</sup><sup>2</sup> <sup>i</sup> ð Þt L 00 <sup>0</sup>, <sup>2</sup> sj � � <sup>þ</sup> <sup>U</sup><sup>n</sup> xxx,i ð Þ<sup>t</sup> <sup>H</sup><sup>3</sup> <sup>i</sup> ð Þt L 00 <sup>0</sup>, <sup>3</sup> sj � � h þU<sup>n</sup> <sup>i</sup>þ<sup>1</sup>ð Þ<sup>t</sup> <sup>L</sup> 00 <sup>1</sup>, <sup>0</sup> sj � � <sup>þ</sup> <sup>U</sup><sup>n</sup> x,iþ<sup>1</sup>Hið Þ<sup>t</sup> <sup>L</sup> 00 <sup>1</sup>,<sup>1</sup> sj � � <sup>þ</sup> <sup>U</sup><sup>n</sup> xx,iþ<sup>1</sup>Hið Þ<sup>t</sup> <sup>L</sup> 00 <sup>1</sup>, <sup>1</sup> sj � � <sup>þ</sup> <sup>U</sup><sup>n</sup> xxx,iþ<sup>1</sup>Hið Þ<sup>t</sup> <sup>L</sup> 00 <sup>1</sup>, <sup>1</sup> sj � �� � <sup>δ</sup><sup>t</sup> 2H<sup>4</sup> i Un <sup>i</sup> ð Þ<sup>t</sup> <sup>L</sup>ð Þ iv <sup>0</sup>, <sup>0</sup> sj � � <sup>þ</sup> <sup>U</sup><sup>n</sup> x,i Hið Þ<sup>t</sup> <sup>L</sup>ð Þ iv <sup>0</sup>,<sup>1</sup> sj � � <sup>þ</sup> <sup>U</sup><sup>n</sup> xx,i ð Þ<sup>t</sup> <sup>H</sup><sup>2</sup> <sup>i</sup> ð Þ<sup>t</sup> <sup>L</sup>ð Þ iv <sup>0</sup>,<sup>2</sup> sj � � <sup>þ</sup> <sup>U</sup><sup>n</sup> xxx,i ð Þ<sup>t</sup> <sup>H</sup><sup>3</sup> <sup>i</sup> ð Þ<sup>t</sup> Liv <sup>0</sup>,<sup>3</sup> sj � � h þU<sup>n</sup> <sup>i</sup>þ<sup>1</sup>ð Þ<sup>t</sup> <sup>L</sup>ð Þ iv <sup>1</sup>, <sup>0</sup> sj � � <sup>þ</sup> <sup>U</sup><sup>n</sup> x,iþ<sup>1</sup>Hið Þ<sup>t</sup> <sup>L</sup>ð Þ iv <sup>1</sup>, <sup>1</sup> sj � � <sup>þ</sup> <sup>U</sup><sup>n</sup> xx,iþ<sup>1</sup>Hið Þ<sup>t</sup> <sup>L</sup>ð Þ iv <sup>1</sup>, <sup>1</sup> sj � � <sup>þ</sup> <sup>U</sup><sup>n</sup> xxx,iþ<sup>1</sup>Hið Þ<sup>t</sup> <sup>L</sup>ð Þ iv <sup>1</sup>,<sup>1</sup> sj � �� (19)

and

β ð Þi <sup>j</sup><sup>1</sup> ¼ L0,<sup>0</sup> sj <sup>þ</sup> δt <sup>2</sup> <sup>U</sup><sup>n</sup> x,i L0,<sup>0</sup> sj <sup>þ</sup> δt 2Hið Þt Un i L0 <sup>0</sup>,<sup>0</sup> sj <sup>þ</sup> δt 2H<sup>2</sup> <sup>i</sup> ð Þt L00 <sup>0</sup>, <sup>0</sup> sj <sup>þ</sup> δt 2H<sup>2</sup> <sup>i</sup> ð Þt Lð Þ iv <sup>0</sup>,<sup>0</sup> sj β ð Þi <sup>j</sup><sup>2</sup> ¼ Hið Þt L0,<sup>1</sup> sj <sup>þ</sup> δt <sup>2</sup> <sup>U</sup><sup>n</sup> x,i Hið Þt L0, <sup>1</sup> sj <sup>þ</sup> δt 2Hið Þt Un <sup>i</sup> Hið Þt L<sup>0</sup> <sup>0</sup>, <sup>1</sup> sj <sup>þ</sup> δt 2H<sup>2</sup> <sup>i</sup> ð Þt Hið Þt L<sup>00</sup> <sup>0</sup>,<sup>1</sup> sj <sup>þ</sup> δt 2H<sup>2</sup> <sup>i</sup> ð Þt Hið Þ<sup>t</sup> <sup>L</sup>ð Þ iv <sup>0</sup>,<sup>1</sup> sj β ð Þi <sup>j</sup><sup>3</sup> <sup>¼</sup> <sup>H</sup><sup>2</sup> <sup>i</sup> ð Þt L0,<sup>2</sup> sj <sup>þ</sup> δt <sup>2</sup> <sup>U</sup><sup>n</sup> x,i H2 <sup>i</sup> ð Þt L0, <sup>2</sup> sj <sup>þ</sup> δt 2Hið Þt Un <sup>i</sup> H<sup>2</sup> <sup>i</sup> ð Þt L<sup>0</sup> <sup>0</sup>,<sup>2</sup> sj <sup>þ</sup> δt 2H<sup>2</sup> <sup>i</sup> ð Þt H2 <sup>i</sup> ð Þt L<sup>00</sup> <sup>0</sup>,<sup>2</sup> sj <sup>þ</sup> δt 2H<sup>4</sup> <sup>i</sup> ð Þt H2 <sup>i</sup> ð Þ<sup>t</sup> <sup>L</sup>ð Þ iv <sup>0</sup>,<sup>2</sup> sj β ð Þi <sup>j</sup><sup>4</sup> <sup>¼</sup> <sup>H</sup><sup>3</sup> <sup>i</sup> ð Þt L0,<sup>3</sup> sj <sup>þ</sup> δt <sup>2</sup> <sup>U</sup><sup>n</sup> x,i H3 <sup>i</sup> ð Þt L0, <sup>3</sup> sj <sup>þ</sup> δt 2Hið Þt Un <sup>i</sup> H<sup>3</sup> <sup>i</sup> ð Þt L<sup>0</sup> <sup>0</sup>,<sup>3</sup> sj <sup>þ</sup> δt 2H<sup>2</sup> <sup>i</sup> ð Þt H3 <sup>i</sup> ð Þt L<sup>00</sup> <sup>0</sup>,<sup>3</sup> sj <sup>þ</sup> δt 2H<sup>4</sup> <sup>i</sup> ð Þt H3 <sup>i</sup> ð Þ<sup>t</sup> <sup>L</sup>ð Þ iv <sup>0</sup>,<sup>3</sup> sj β ð Þi <sup>j</sup><sup>5</sup> ¼ L1,<sup>0</sup> sj <sup>þ</sup> δt <sup>2</sup> <sup>U</sup><sup>n</sup> x,i L1,<sup>0</sup> sj <sup>þ</sup> δt 2Hið Þt Un i L0 <sup>1</sup>,<sup>0</sup> sj <sup>þ</sup> δt 2H<sup>2</sup> <sup>i</sup> ð Þt L00 <sup>1</sup>, <sup>0</sup> sj <sup>þ</sup> δt 2H<sup>4</sup> <sup>i</sup> ð Þt Lð Þ iv <sup>1</sup>,<sup>0</sup> sj β ð Þi <sup>j</sup><sup>6</sup> ¼ Hið Þt L1, <sup>1</sup> sj <sup>þ</sup> δt <sup>2</sup> <sup>U</sup><sup>n</sup> x,i Hið Þt L1, <sup>1</sup> sj <sup>þ</sup> δt 2Hið Þt Un <sup>i</sup> Hið Þt L<sup>0</sup> <sup>1</sup>,<sup>1</sup> sj <sup>þ</sup> δt 2H<sup>2</sup> <sup>i</sup> ð Þt Hið Þt L<sup>00</sup> <sup>1</sup>,<sup>1</sup> sj <sup>þ</sup> δt 2H<sup>4</sup> <sup>i</sup> ð Þt Hið Þ<sup>t</sup> <sup>L</sup>ð Þ iv <sup>1</sup>,<sup>1</sup> sj β ð Þi <sup>j</sup><sup>7</sup> <sup>¼</sup> <sup>H</sup><sup>2</sup> <sup>i</sup> ð Þt L1,<sup>2</sup> sj <sup>þ</sup> δt <sup>2</sup> <sup>U</sup><sup>n</sup> x,i H2 <sup>i</sup> ð Þt L1, <sup>2</sup> sj <sup>þ</sup> δt 2Hið Þt Un <sup>i</sup> H<sup>2</sup> <sup>i</sup> ð Þt L<sup>0</sup> <sup>1</sup>,<sup>2</sup> sj <sup>þ</sup> δt 2H<sup>2</sup> <sup>i</sup> ð Þt H2 <sup>i</sup> ð Þt L<sup>00</sup> <sup>1</sup>,<sup>2</sup> sj <sup>þ</sup> δt 2H<sup>4</sup> <sup>i</sup> ð Þt H2 <sup>i</sup> ð Þ<sup>t</sup> <sup>L</sup>ð Þ iv <sup>1</sup>,<sup>2</sup> sj β ð Þi <sup>j</sup><sup>8</sup> <sup>¼</sup> <sup>H</sup><sup>3</sup> <sup>i</sup> ð Þt L1,<sup>3</sup> sj <sup>þ</sup> δt <sup>2</sup> <sup>U</sup><sup>n</sup> x,i H3 <sup>i</sup> ð Þt L1, <sup>3</sup> sj <sup>þ</sup> δt 2Hið Þt Un <sup>i</sup> H<sup>3</sup> <sup>i</sup> ð Þt L<sup>0</sup> <sup>1</sup>,<sup>3</sup> sj <sup>þ</sup> δt 2H<sup>2</sup> <sup>i</sup> ð Þt H3 <sup>i</sup> ð Þt L<sup>00</sup> <sup>1</sup>,<sup>3</sup> sj <sup>þ</sup> δt 2H<sup>4</sup> <sup>i</sup> ð Þt H3 <sup>i</sup> ð Þ<sup>t</sup> <sup>L</sup>ð Þ iv <sup>1</sup>,<sup>3</sup> sj (20)

From the boundary conditions (28) and (29), one gets

$$\begin{aligned} \mathcal{U}(\mathbf{x}\_1) &= \sigma \\ \mathcal{U}\_\mathbf{x}(\mathbf{x}\_1) &= \beta \\ \mathcal{U}(\mathbf{x}\_{N+1}) &= \omega \\ \mathcal{U}\_\mathbf{x}(\mathbf{x}\_{N+1}) &= \zeta \end{aligned} \tag{21}$$

6. Solution adjustment by interpolation

where Xn

mesh <sup>X</sup><sup>~</sup> <sup>n</sup> i n o<sup>N</sup>þ<sup>1</sup>

Xn i � �<sup>N</sup>þ<sup>1</sup>

<sup>i</sup> <sup>¼</sup> Xið Þ tn with <sup>H</sup><sup>n</sup>

derivatives given by U<sup>n</sup>

new approximations <sup>U</sup><sup>~</sup> <sup>n</sup>

Discretization of the time domain ta ½ � ; tb is done using the following finite sequence

At each time <sup>t</sup> <sup>¼</sup> tn <sup>¼</sup> <sup>n</sup> � dt, consider a non-uniform spatial mesh <sup>X</sup><sup>n</sup>

<sup>i</sup>þ<sup>1</sup> � Xn

<sup>i</sup>¼<sup>1</sup> and <sup>U</sup>ð Þ<sup>l</sup>

and <sup>U</sup><sup>~</sup> ð Þ<sup>l</sup> i n o � �<sup>n</sup> <sup>N</sup>þ<sup>1</sup>

<sup>i</sup> <sup>¼</sup> <sup>X</sup><sup>n</sup>

i � �<sup>N</sup>þ<sup>1</sup>

i n o<sup>N</sup>þ<sup>1</sup> i¼1

in a similar manner the approximations U<sup>n</sup>

polynomial (14) is written in compact form as

where the 4ð Þ N þ 1 unknowns are given by

Uð Þ<sup>i</sup> <sup>i</sup> <sup>¼</sup> <sup>∂</sup><sup>l</sup> u

of <sup>U</sup>ð Þ<sup>l</sup> ð Þ<sup>x</sup> is required at <sup>x</sup> <sup>¼</sup> <sup>X</sup>~<sup>i</sup>

<sup>n</sup> � � is then defined as

<sup>n</sup> is defined as

nate s of X~<sup>i</sup>

U~ ð Þ<sup>l</sup> X~ <sup>i</sup>

X<sup>3</sup>

Given the partition (23) and approximations Uð Þ<sup>l</sup>

<sup>l</sup>¼<sup>0</sup> ð Þ hl ð Þ<sup>l</sup> <sup>U</sup>ð Þ<sup>l</sup>

<sup>∂</sup>xl ð Þ Xið Þ<sup>t</sup> ; <sup>t</sup> <sup>U</sup>ð Þ<sup>l</sup>

<sup>n</sup> where X~<sup>i</sup>

<sup>a</sup> <sup>¼</sup> Xn

i n o � �<sup>n</sup> <sup>N</sup>þ<sup>1</sup>

<sup>1</sup> < … < X<sup>n</sup>

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

same time step t ¼ tn one also considers the approximations to the exact solution u xð Þ ; t and its

derivative approximation with respect to the variable x at the time t ¼ tn For l ¼ 1, 2, 3. A new

i¼1

i � �<sup>N</sup>þ<sup>1</sup>

<sup>i</sup> <sup>L</sup>0,lð Þþ <sup>s</sup> <sup>X</sup><sup>3</sup>

<sup>i</sup>þ<sup>1</sup> <sup>≈</sup> <sup>∂</sup><sup>l</sup> u

> <sup>n</sup> ∈ Xn <sup>i</sup> ; X<sup>n</sup> iþ1

<sup>s</sup> <sup>¼</sup> <sup>X</sup>~<sup>i</sup>

<sup>i</sup>¼<sup>1</sup> is generated by (2) at each current time step tn. The goal is to determine the

<sup>i</sup>¼<sup>1</sup> and <sup>U</sup>ð Þ<sup>l</sup>

<sup>l</sup>¼<sup>0</sup> ð Þ hl

i n o � �<sup>n</sup>

> <sup>n</sup> � <sup>X</sup><sup>n</sup> i Hn i

ð Þ<sup>l</sup> Uð Þ<sup>l</sup>

<sup>∂</sup>xl ð Þ Xiþ<sup>1</sup>ð Þ<sup>t</sup> ; <sup>t</sup> , l <sup>¼</sup> <sup>0</sup>, <sup>1</sup>, <sup>2</sup>, <sup>3</sup>:

<sup>i</sup>¼<sup>1</sup> in each subinterval Xi ½ � ; Xiþ<sup>1</sup> . 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 Xi ½ � ; Xiþ<sup>1</sup> . The Hermite

f g ta ¼ t<sup>0</sup> < … < tn < … < tk ¼ tb (22)

<sup>i</sup>¼<sup>1</sup> respectively where <sup>U</sup>ð Þ<sup>l</sup>

i n o � �<sup>n</sup> <sup>N</sup>þ<sup>1</sup>

i � �<sup>N</sup>þ<sup>1</sup>

<sup>i</sup> being a non-uniform spatial step for i ¼ 1, …, N. At the

<sup>N</sup>þ<sup>1</sup> <sup>¼</sup> <sup>b</sup> (23)

i � �<sup>n</sup>

which are related to the new mesh <sup>X</sup><sup>~</sup> <sup>n</sup>

i¼1

<sup>i</sup>¼<sup>1</sup> given by

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

represents the l

are related to the old mesh

<sup>i</sup>þ<sup>1</sup>L1,lð Þ<sup>s</sup> (24)

for l ¼ 0, 1, 2, 3, suppose interpolation

� � for <sup>i</sup> <sup>¼</sup> <sup>1</sup>, …, N. Firstly, the local coordi-

i n o<sup>N</sup>þ<sup>1</sup> i¼1

(25)

th

269

which results in a consistent system of 4N þ 4 equations in 4N þ 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:

