6. Solution adjustment by interpolation

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

> <sup>þ</sup> δt <sup>2</sup> <sup>U</sup><sup>n</sup> x,i

<sup>j</sup><sup>2</sup> ¼ Hið Þt L0,<sup>1</sup> sj

<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>

<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>

> β ð Þi <sup>j</sup><sup>5</sup> ¼ L1,<sup>0</sup> sj

<sup>j</sup><sup>6</sup> ¼ Hið Þt L1, <sup>1</sup> sj

<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>

<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>

approach is as follows:

β ð Þi

β ð Þi <sup>j</sup><sup>3</sup> <sup>¼</sup> <sup>H</sup><sup>2</sup>

β ð Þi <sup>j</sup><sup>4</sup> <sup>¼</sup> <sup>H</sup><sup>3</sup>

> β ð Þi

β ð Þi <sup>j</sup><sup>7</sup> <sup>¼</sup> <sup>H</sup><sup>2</sup>

β ð Þi <sup>j</sup><sup>8</sup> <sup>¼</sup> <sup>H</sup><sup>3</sup>  <sup>þ</sup> δt <sup>2</sup> <sup>U</sup><sup>n</sup> x,i L0,<sup>0</sup> sj <sup>þ</sup>

 <sup>þ</sup> δt <sup>2</sup> <sup>U</sup><sup>n</sup> x,i L1,<sup>0</sup> sj <sup>þ</sup>

Hið Þt L0, <sup>1</sup> sj

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

Hið Þt L1, <sup>1</sup> sj

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

5. Solution approach for the PDE system

1. Solve the given physical PDE on the current mesh.

3. Find the new mesh by solving a MMPDE.

2. Use the PDE solution obtained to calculate the monitor function.

4. Adjust the current PDE solution to suite the new mesh by interpolation. 5. Solve the physical PDE on the new mesh for the solution in the next time.

<sup>þ</sup>

<sup>þ</sup>

δt 2Hið Þt Un i L0 <sup>0</sup>,<sup>0</sup> sj <sup>þ</sup>

δt 2Hið Þt Un <sup>i</sup> Hið Þt L<sup>0</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 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 2Hið Þt Un <sup>i</sup> Hið Þt L<sup>0</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 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>

which results in a consistent system of 4N þ 4 equations in 4N þ 4 unknowns.

U xð Þ¼ <sup>1</sup> σ Uxð Þ¼ x<sup>1</sup> β U xð Þ¼ <sup>N</sup>þ<sup>1</sup> ω Uxð Þ¼ xNþ<sup>1</sup> ζ

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

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>0</sup>, <sup>0</sup> sj <sup>þ</sup>

> δt 2H<sup>2</sup> <sup>i</sup> ð Þt

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>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>2</sup> <sup>i</sup> ð Þt

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>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>2</sup> <sup>i</sup> ð Þt L00 <sup>1</sup>, <sup>0</sup> sj <sup>þ</sup>

Hið Þt L<sup>00</sup> <sup>0</sup>,<sup>1</sup> sj <sup>þ</sup>

Hið Þt L<sup>00</sup> <sup>1</sup>,<sup>1</sup> sj <sup>þ</sup>

<sup>0</sup>, <sup>1</sup> sj <sup>þ</sup>

<sup>1</sup>,<sup>1</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 

δt 2H<sup>4</sup> <sup>i</sup> ð Þt Lð Þ iv <sup>1</sup>,<sup>0</sup> sj 

δt 2H<sup>2</sup> <sup>i</sup> ð Þt

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 

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 

δt 2H<sup>4</sup> <sup>i</sup> ð Þt

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 

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 

Hið Þ<sup>t</sup> <sup>L</sup>ð Þ iv <sup>0</sup>,<sup>1</sup> sj 

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

(20)

(21)

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

$$\left\{ t\_{\mathfrak{t}} = t\_0 < \dots < t\_{\mathfrak{t}} < \dots < t\_k = t\_{\mathfrak{t}} \right\} \tag{22}$$

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> i � �<sup>N</sup>þ<sup>1</sup> <sup>i</sup>¼<sup>1</sup> given by

$$a = X\_1^n < \dots < X\_{N+1}^n = b \tag{23}$$

where Xn <sup>i</sup> <sup>¼</sup> Xið Þ tn with <sup>H</sup><sup>n</sup> <sup>i</sup> <sup>¼</sup> <sup>X</sup><sup>n</sup> <sup>i</sup>þ<sup>1</sup> � Xn <sup>i</sup> being a non-uniform spatial step for i ¼ 1, …, N. At the same time step t ¼ tn one also considers the approximations to the exact solution u xð Þ ; t and its derivatives given by U<sup>n</sup> i � �<sup>N</sup>þ<sup>1</sup> <sup>i</sup>¼<sup>1</sup> and <sup>U</sup>ð Þ<sup>l</sup> i n o � �<sup>n</sup> <sup>N</sup>þ<sup>1</sup> <sup>i</sup>¼<sup>1</sup> respectively where <sup>U</sup>ð Þ<sup>l</sup> i � �<sup>n</sup> represents the l th derivative approximation with respect to the variable x at the time t ¼ tn For l ¼ 1, 2, 3. A new mesh <sup>X</sup><sup>~</sup> <sup>n</sup> i n o<sup>N</sup>þ<sup>1</sup> <sup>i</sup>¼<sup>1</sup> is generated by (2) at each current time step tn. The goal is to determine the new approximations <sup>U</sup><sup>~</sup> <sup>n</sup> i n o<sup>N</sup>þ<sup>1</sup> i¼1 and <sup>U</sup><sup>~</sup> ð Þ<sup>l</sup> i n o � �<sup>n</sup> <sup>N</sup>þ<sup>1</sup> i¼1 which are related to the new mesh <sup>X</sup><sup>~</sup> <sup>n</sup> i n o<sup>N</sup>þ<sup>1</sup> i¼1 in a similar manner the approximations U<sup>n</sup> i � �<sup>N</sup>þ<sup>1</sup> <sup>i</sup>¼<sup>1</sup> and <sup>U</sup>ð Þ<sup>l</sup> i n o � �<sup>n</sup> <sup>N</sup>þ<sup>1</sup> i¼1 are related to the old mesh Xn i � �<sup>N</sup>þ<sup>1</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 polynomial (14) is written in compact form as

$$\sum\_{l=0}^{3} \binom{3}{l} {}^{(l)} \mathcal{U}\_{i}^{(l)} L\_{0,l}(\mathbf{s}) + \sum\_{l=0}^{3} {}^{3} \left( h \right) {}^{(l)} \mathcal{U}\_{i+1}^{(l)} L\_{1,l}(\mathbf{s}) \tag{24}$$

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

$$\mathcal{U}\_{l}^{(l)} = \frac{\partial^l \mathcal{u}}{\partial \mathbf{x}^l}(\mathcal{X}\_l(t), t) \mathcal{U}\_{i+1}^{(l)} \approx \frac{\partial^l \mathcal{u}}{\partial \mathbf{x}^l}(\mathcal{X}\_{i+1}(t), t), \quad l = 0, 1, 2, 3.$$

Given the partition (23) and approximations Uð Þ<sup>l</sup> i n o � �<sup>n</sup> for l ¼ 0, 1, 2, 3, suppose interpolation 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> where X~<sup>i</sup> <sup>n</sup> ∈ Xn <sup>i</sup> ; X<sup>n</sup> iþ1 � � for <sup>i</sup> <sup>¼</sup> <sup>1</sup>, …, N. Firstly, the local coordinate s of X~<sup>i</sup> <sup>n</sup> is defined as

$$s = \frac{\check{X}\_i^{\;n} - X\_i^n}{H\_i^n} \tag{25}$$

U~ ð Þ<sup>l</sup> X~ <sup>i</sup> <sup>n</sup> � � is then defined as

$$\tilde{\mathbf{U}}^{(l)}\left(\tilde{\mathbf{X}}\_{i}^{\n}\right) = \sum\_{i=0}^{3} H\_{i}^{l-p} \mathbf{U}\_{i}^{(l)} \frac{d^{(l)} \mathbf{L}\_{0,l}(\mathbf{s})}{d\mathbf{s}^{(l)}} + \sum\_{i=0}^{3} H\_{i}^{l-p} \mathbf{U}\_{i+1}^{(l)} \frac{d^{(l)} \mathbf{L}\_{0,l}(\mathbf{s})}{d\mathbf{s}^{(l)}} \tag{26}$$

for <sup>l</sup> <sup>¼</sup> <sup>0</sup>, <sup>1</sup>, <sup>2</sup>, 3 to give the interpolated values of <sup>U</sup><sup>~</sup> and the first three consecutive derivatives on the new subinterval <sup>X</sup><sup>~</sup> <sup>n</sup> <sup>i</sup> ; <sup>X</sup><sup>~</sup> <sup>n</sup> iþ1 h i. In order to compute the approximations of <sup>U</sup> at the next time step <sup>t</sup> <sup>¼</sup> tnþ<sup>1</sup> denoted by <sup>U</sup><sup>n</sup> i � �<sup>N</sup>þ<sup>1</sup> <sup>i</sup>¼<sup>1</sup> , the values of the new mesh <sup>X</sup><sup>~</sup> <sup>n</sup> i n o<sup>N</sup>þ<sup>1</sup> <sup>i</sup>¼<sup>1</sup> and the updated approximations <sup>U</sup><sup>~</sup> <sup>n</sup> i n o<sup>N</sup>þ<sup>1</sup> <sup>i</sup>¼<sup>1</sup> are used in a septic Hermite collocation numerical scheme. The new approximations U<sup>n</sup>þ<sup>1</sup> i � �<sup>N</sup>þ<sup>1</sup> <sup>i</sup>¼<sup>1</sup> and the new mesh <sup>X</sup><sup>~</sup> <sup>n</sup>þ<sup>1</sup> i n o<sup>N</sup>þ<sup>1</sup> <sup>i</sup>¼<sup>1</sup> become the starting conditions for repeating the whole adaptive process.

### 7. Numerical results

Consider the KSe

$$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + \frac{\partial^2 u}{\partial x^2} + \frac{\partial^4 u}{\partial x^4} = 0, \qquad t > 0 \tag{27}$$

−30 −20 −10 0 10 20 30

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numerical solution exact solution

271

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x

−30 −20 −10 <sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>0</sup>

x

Figure 2. Hermite collocation method, uniform mesh, absolute error in numerical solution of KSe at t ¼ 4, N ¼ 100 and

Figure 1. Hermite collocation method, uniform mesh, numerical solution behaviour of KSe at t ¼ 4 with N ¼ 100 and

3.5

3

0.005

0.01

0.015

0.02

Absolute Error

0.025

0.03

0.035

0.04

δt ¼ 0:001.

δt ¼ 0:001.

4

4.5

5

u

5.5

6

6.5

7

in the domain ½ � �30; 30 , t > 0 with boundary conditions

$$u(-\mathfrak{A}0,t) = \sigma, \qquad \qquad u\_x(-\mathfrak{A}0,t) = \beta \tag{28}$$

$$u(\mathfrak{A}0,t) = \omega,\tag{20} \llcorner \omega \llcorner (\mathfrak{A}0,t) = \zeta \tag{29}$$

Where σ, β, ω and ζ are obtained from the exact solution

$$u(\mathbf{x},t) = c + \frac{15}{19} \sqrt{\frac{11}{19}} [-9 \tanh^3(k(\mathbf{x} - ct - \mathbf{x}\_0)) + 11 \tanh(k(\mathbf{x} - ct - \mathbf{x}\_0))] \tag{30}$$

With <sup>c</sup> <sup>¼</sup> <sup>0</sup>:1, x<sup>0</sup> ¼ �12 and <sup>k</sup> <sup>¼</sup> <sup>1</sup> 2 ffiffiffiffi 11 19 q .

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 t ¼ 4 with N ¼ 100 and δt ¼ 0:001. In Figure 1, one observes that the numerical solution tracks the exact solution with the absolute error variation as shown in Figure 2.

Figure 3 shows the solution obtained by the collocation method on a stationary mesh for time t ¼ 0, 1, 2, 3, 4. The movement of the solution is from left to right as time increases and the solution tracks the exact solution with no oscillations. One also observes that the concentration of mesh points is higher in the flatter regions of the solution profile in comparison to the concentration in the steeper region.

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U~ ð Þ<sup>l</sup> X~ <sup>i</sup> <sup>n</sup> � �

on the new subinterval <sup>X</sup><sup>~</sup> <sup>n</sup>

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

approximations U<sup>n</sup>þ<sup>1</sup>

7. Numerical results

Consider the KSe

time step <sup>t</sup> <sup>¼</sup> tnþ<sup>1</sup> denoted by <sup>U</sup><sup>n</sup>

i n o<sup>N</sup>þ<sup>1</sup>

i � �<sup>N</sup>þ<sup>1</sup>

repeating the whole adaptive process.

<sup>¼</sup> <sup>X</sup> 3

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

<sup>i</sup> ; <sup>X</sup><sup>~</sup> <sup>n</sup> iþ1 h i

i¼0

i � �<sup>N</sup>þ<sup>1</sup>

∂u ∂t þ u ∂u ∂x þ ∂<sup>2</sup>u ∂x<sup>2</sup> þ

in the domain ½ � �30; 30 , t > 0 with boundary conditions

Where σ, β, ω and ζ are obtained from the exact solution

ffiffiffiffiffi 11 19 r

> 2 ffiffiffiffi 11 19 q .

solution with the absolute error variation as shown in Figure 2.

�9 tanh<sup>3</sup>

15 19

u xð Þ¼ ; t c þ

With <sup>c</sup> <sup>¼</sup> <sup>0</sup>:1, x<sup>0</sup> ¼ �12 and <sup>k</sup> <sup>¼</sup> <sup>1</sup>

concentration in the steeper region.

<sup>i</sup>¼<sup>1</sup> and the new mesh <sup>X</sup><sup>~</sup> <sup>n</sup>þ<sup>1</sup>

H<sup>l</sup>�<sup>p</sup> <sup>i</sup> <sup>U</sup>ð Þ<sup>l</sup> i

<sup>d</sup>ð Þ<sup>l</sup> <sup>L</sup>0,lð Þ<sup>s</sup> dsð Þ<sup>l</sup> <sup>þ</sup><sup>X</sup>

for <sup>l</sup> <sup>¼</sup> <sup>0</sup>, <sup>1</sup>, <sup>2</sup>, 3 to give the interpolated values of <sup>U</sup><sup>~</sup> and the first three consecutive derivatives

3

H<sup>l</sup>�<sup>p</sup> <sup>i</sup> <sup>U</sup>ð Þ<sup>l</sup> iþ1

. In order to compute the approximations of U at the next

<sup>d</sup>ð Þ<sup>l</sup> <sup>L</sup>0,lð Þ<sup>s</sup>

i n o<sup>N</sup>þ<sup>1</sup>

<sup>∂</sup>x<sup>4</sup> <sup>¼</sup> <sup>0</sup>, t <sup>&</sup>gt; <sup>0</sup> (27)

<sup>i</sup>¼<sup>1</sup> become the starting conditions for

dsð Þ<sup>l</sup> (26)

<sup>i</sup>¼<sup>1</sup> and the updated

i¼0

<sup>i</sup>¼<sup>1</sup> , the values of the new mesh <sup>X</sup><sup>~</sup> <sup>n</sup>

i n o<sup>N</sup>þ<sup>1</sup>

∂<sup>4</sup>u

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 t ¼ 4 with N ¼ 100 and δt ¼ 0:001. In Figure 1, one observes that the numerical solution tracks the exact

Figure 3 shows the solution obtained by the collocation method on a stationary mesh for time t ¼ 0, 1, 2, 3, 4. The movement of the solution is from left to right as time increases and the solution tracks the exact solution with no oscillations. One also observes that the concentration of mesh points is higher in the flatter regions of the solution profile in comparison to the

<sup>i</sup>¼<sup>1</sup> are used in a septic Hermite collocation numerical scheme. The new

uð Þ¼ �30; t σ, uxð Þ¼ �30; t β (28)

uð Þ¼ 30; t ω, uxð Þ¼ 30; t ζ (29)

ðk xð Þ � ct � x<sup>0</sup> Þ þ 11 tanhð Þ k xð Þ � ct � x<sup>0</sup> � � (30)

Figure 1. Hermite collocation method, uniform mesh, numerical solution behaviour of KSe at t ¼ 4 with N ¼ 100 and δt ¼ 0:001.

Figure 2. Hermite collocation method, uniform mesh, absolute error in numerical solution of KSe at t ¼ 4, N ¼ 100 and δt ¼ 0:001.

Figure 3. Hermite collocation method, stationary mesh, numerical solution behaviour of KSe problem with N ¼ 100, δt ¼ 0:001 up to final time T ¼ 4:

Figure 6 shows the numerical solution profiles produced by the adaptive collocation method for time t ¼ 0, 1, 2, 3, 4. One observes that the solution moves from left to right as time progresses. The mesh points at different times keep on tracking the solution profile and maintain an almost equal distribution along the profile up to final time T ¼ 4. Figure 7 shows the paths taken by the mesh points in tracking the solution profile. In Table 1, the infinity norm error for an adaptive collocation method is calculated and results are compared with the method in [13]. Results show improvements in the maximum point wise errors when an adaptive Hermite

−30 −20 −10 0 10 20 30

numerical solution exact solution

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

273

x

Figure 6. Hermite collocation method, adaptive mesh, numerical solution behaviour of KSe up to final time T ¼ 4 for

Figure 5. Hermite collocation method, non-uniform mesh, absolute error in numerical solution of KSe at t ¼ 100, δt ¼ 0:001,

−30 −20 −10 <sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>0</sup>

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

x

collocation method is used.

3.5 4 4.5 5 5.5 6 6.5 7

3

<sup>N</sup> <sup>¼</sup> <sup>100</sup>, <sup>δ</sup><sup>t</sup> <sup>¼</sup> <sup>0</sup>:001, <sup>τ</sup> <sup>¼</sup> <sup>2</sup> � <sup>10</sup>�<sup>2</sup> and <sup>α</sup> <sup>¼</sup> 8.

u

<sup>τ</sup> <sup>¼</sup> <sup>2</sup> � <sup>10</sup>�<sup>2</sup> and <sup>α</sup> <sup>¼</sup> 8.

0.5

1

1.5

Absolute Error

2

2.5 x 10−3

Figures 4 and 5 show the numerical solution profile and the behaviour of the maximum absolute error, respectively at t ¼ 4 with N ¼ 100, δt ¼ 0:001 and α ¼ 8 on an adaptive mesh. In Figure 4, one observes that the numerical solution is able to track the exact solution and the distribution of mesh points is almost equal along the solution profile which enables resolution of the solution with minimum errors.

Figure 4. Hermite collocation method, non-uniform mesh, numerical solution behaviour of KSe problem at t ¼ 4 with <sup>N</sup> <sup>¼</sup> <sup>100</sup>, <sup>δ</sup><sup>t</sup> <sup>¼</sup> <sup>0</sup>:001, <sup>τ</sup> <sup>¼</sup> <sup>2</sup> � <sup>10</sup>�<sup>2</sup> and <sup>α</sup> <sup>¼</sup> 8.

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Figure 5. Hermite collocation method, non-uniform mesh, absolute error in numerical solution of KSe at t ¼ 100, δt ¼ 0:001, <sup>τ</sup> <sup>¼</sup> <sup>2</sup> � <sup>10</sup>�<sup>2</sup> and <sup>α</sup> <sup>¼</sup> 8.

Figure 6 shows the numerical solution profiles produced by the adaptive collocation method for time t ¼ 0, 1, 2, 3, 4. One observes that the solution moves from left to right as time progresses. The mesh points at different times keep on tracking the solution profile and maintain an almost equal distribution along the profile up to final time T ¼ 4. Figure 7 shows the paths taken by the mesh points in tracking the solution profile. In Table 1, the infinity norm error for an adaptive collocation method is calculated and results are compared with the method in [13]. Results show improvements in the maximum point wise errors when an adaptive Hermite collocation method is used.

Figures 4 and 5 show the numerical solution profile and the behaviour of the maximum absolute error, respectively at t ¼ 4 with N ¼ 100, δt ¼ 0:001 and α ¼ 8 on an adaptive mesh. In Figure 4, one observes that the numerical solution is able to track the exact solution and the distribution of mesh points is almost equal along the solution profile which enables resolution

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x

Figure 4. Hermite collocation method, non-uniform mesh, numerical solution behaviour of KSe problem at t ¼ 4 with

Figure 3. Hermite collocation method, stationary mesh, numerical solution behaviour of KSe problem with N ¼ 100,

−30 −20 −10 0 10 20 30

numerical solution exact solution

numerical solution exact solution

x

of the solution with minimum errors.

3.5

3

<sup>N</sup> <sup>¼</sup> <sup>100</sup>, <sup>δ</sup><sup>t</sup> <sup>¼</sup> <sup>0</sup>:001, <sup>τ</sup> <sup>¼</sup> <sup>2</sup> � <sup>10</sup>�<sup>2</sup> and <sup>α</sup> <sup>¼</sup> 8.

4

4.5

5

u

5.5

6

6.5

7

3.5

3

δt ¼ 0:001 up to final time T ¼ 4:

4

4.5

5

u

5.5

6

6.5

7

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

Figure 6. Hermite collocation method, adaptive mesh, numerical solution behaviour of KSe up to final time T ¼ 4 for <sup>N</sup> <sup>¼</sup> <sup>100</sup>, <sup>δ</sup><sup>t</sup> <sup>¼</sup> <sup>0</sup>:001, <sup>τ</sup> <sup>¼</sup> <sup>2</sup> � <sup>10</sup>�<sup>2</sup> and <sup>α</sup> <sup>¼</sup> 8.

Author details

References

Denson Muzadziwa1

, Stephen T. Sikwila<sup>2</sup> and Stanford Shateyi<sup>3</sup>

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

[1] Sivashinsky GI. Nonlinear analysis of hydrodynamic instability in laminar flames-I-der-

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[3] Kuramoto Y. Instability and turbulence of wavefronts in reaction-diffusion systems. Pro-

[4] Kuramoto Y, Tsuzuki T. Diffusion-induced chaos in reaction systems. Progress of Theo-

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[7] Sivashinsky GI, Michelson D. On irregular wavy flow of a liquid film down a vertical

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[9] Mittal RC, Arora G. Quintic b-spline collocation method for numerical solution of the Kuramoto-Sivashinsky equation. Communications in Nonlinear Science and Numerical

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ivation of basic equations. Acta Astronautica. 1977;4:1177-1206

\*Address all correspondence to: stanford.shateyi@univen.ac.za

1 University of Zimbabwe, Harare, Zimbabwe 2 Sol Plaatje University, Kimberley, South Africa

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3 University of Venda, Thohoyandou, South Africa

\*

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

275

Figure 7. Hermite collocation method, mesh trajectories of KSe equation up to final time T ¼ 4 with N ¼ 100, δt ¼ 0:001, <sup>τ</sup> <sup>¼</sup> <sup>2</sup> � <sup>10</sup>�<sup>2</sup> and <sup>α</sup> <sup>¼</sup> 8.


Table 1. Comparison of maximum pointwise errors in the numerical solution of the KSe on an adaptive mesh at different times with δt ¼ 0:001 and N ¼ 100.

### 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.
