Algorithm for Computation

The solution to the finite difference scheme is obtained through recursive back-substitution.

Equation 31 - is solved by,

$$\begin{eqnarray} A_{ij}^{n+\frac{1}{2}}= g_{22}\,(\mathbf{s}^n_{ij}) \quad C_{... ...athbf{s}^n_{ij})\;\mathbf{S}_{ij}(u^n)\;+\;\mathbf{P}_{ij}\,(u^n).\end{eqnarray}$$ (43)

The solution for a fixed number j in range $1 \le j \le N-1$obtained through,

$$\begin{eqnarray} u^{n+1/2}_{ij}= \alpha^{n+1/2}_{i+1j}\; u^{n+\frac{1}{2}}_{i+1\... ... i=1,N-1,& \;\;\;\;\;\;\;\; u^{n+\frac{1}{2}}_{Nj} = \phi_1(1,jh),\end{eqnarray}$$ (44)

where,

$$\begin{eqnarray} \alpha^{n+1/2}_{i+1j}= \frac{B_{ij}^{n+\frac{1}{2}}}{C_{ij}^{n+... ...,N-1, & \quad \beta^{n+\frac{1}{2}}_{0j} = \phi_1(0,jh). \nonumber\end{eqnarray}$$

Equation 32 - is solved by,

$$\begin{eqnarray} A_{ij}^{n+\frac{1}{2}}= g_{22}\,(\mathbf{s}^n_{ij}) \quad C_{... ...athbf{s}^n_{ij})\;\mathbf{S}_{ij}(v^n)\;+\;\mathbf{P}_{ij}\,(v^n).\end{eqnarray}$$ (45)

The solution for a fixed j in range $1 \le j \le N-1$ is obtained through,

$$\begin{eqnarray} v^{n+1/2}_{ij}= \alpha^{n+1/2}_{i+1j}\; v^{n+\frac{1}{2}}_{i+1\... ... i=1,N-1,& \;\;\;\;\;\;\;\; v^{n+\frac{1}{2}}_{Nj} = \phi_2(1,jh),\end{eqnarray}$$ (46)

where,

$$\begin{eqnarray} \alpha^{n+1/2}_{i+1j}= \frac{B_{ij}^{n+\frac{1}{2}}}{C_{ij}^{n+... ...1,N-1, & \quad \beta^{n+\frac{1}{2}}_{0j} = \phi_2(0,jh) \nonumber\end{eqnarray}$$

Equation 33 - is solved by,

$$\begin{eqnarray} A_{ij}^{n+\frac{1}{2}}= g_{11}\,(\mathbf{s}^n_{ij}) \quad C_{... ...^{n+1/2}_{ij} - g_{11}\,(\mathbf{s}^n_{ij})\;\mathbf{S}_{ij}(u^n).\end{eqnarray}$$ (47)

The solution for a fixed i in the range $1 \le i \le N-1$ is obtained through,

$$\begin{eqnarray} u^{n+1}_{ij}= \alpha^{n+1}_{ij+1}\; u^{n+1}_{ij+1}+ \beta^{n+1}_{ij+1}, & \quad i=1,N-1,& \quad u^{n+1}_{iN} = \phi_1(ih,1),\end{eqnarray}$$ (48)

where,

$$\begin{eqnarray} \alpha^{n+1}_{i+1j}= \frac{B_{ij}^{n+1}}{C_{ij}^{n+1}- \alpha^{... ...& \quad j=1,N-1, & \quad \beta^{n+1}_{i0} = \phi_1(ih,0) \nonumber\end{eqnarray}$$

Equation 34 - is solved by,

$$\begin{eqnarray} A_{ij}^{n+\frac{1}{2}}= g_{11}\,(\mathbf{s}^n_{ij}) \quad C_{... ...^{n+1/2}_{ij} - g_{11}\,(\mathbf{s}^n_{ij})\;\mathbf{S}_{ij}(v^n).\end{eqnarray}$$ (49)

The solution for a fixed i in the range $1 \le j \le N-1$ is obtained through,

$$\begin{eqnarray} v^{n+1}_{ij}= \alpha^{n+1}_{ij+1}\; v^{n+1}_{ij+1}+ \beta^{n+1}_{ij+1}, & \quad i=1,N-1,& \quad v^{n+1}_{iN} = \phi_2(ih,0),\end{eqnarray}$$ (50)

where,

$$\begin{eqnarray} \alpha^{n+1}_{i+1j}= \frac{B_{ij}^{n+1}}{C_{ij}^{n+1}- \alpha^{... ... \quad j=1,N-1, & \quad \beta^{n+1}_{i0} = \phi_1(ih,0). \nonumber\end{eqnarray}$$

An approximate solution of the equations 19 will be found at some large time T_N that satisfies,  

$$\max_{0\le i,j,\le N} \frac{1}{\tau} \sqrt{ (u^{n+1}_{ij}-u^n_{ij})^{2} + (v^{n+1}_{ij}-v^n_{ij})^{2}} \le \epsilon.$$ (51)