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Solution of Tridiagonal systems:


The implicit difference formula given above has involved three unknown values of U at the advanced time level . The system of linear algebraic equations arising from the implicit difference formulae which must be solved at each time step is a special case of the tridiagonal system

$\displaystyle -\alpha_{m}U_{m-1}+\beta_{m}U_{m}-r_{m}U_{m+1}=b_{m}$

for $ 1\leq m \leq M-1,$ where $ U_{0}$ and $ U_{M.}$ are known from the boundary condition. If

$\displaystyle \alpha_{m}>0, \quad \beta_{m}>0, \quad r_{m}>0$

and $ \beta_{m}\geq (\alpha_{m}+r_{m})\quad 1\leq m\leq M-1$, a highly efficient method is known for solving the tridiagonal system. The method is given as follows:
consider the difference relation

$\displaystyle U_{m}=W_{m}U_{m+1}+ g_{m}$

for $ 0\leq m \leq M-1$, from which it follows that

$\displaystyle U_{m-1}=W_{m-1}U_{m}+ g_{m-1}$

If this is used to eliminate $ U_{m-1}$ from the original difference formula defining the tridiagonal system, the result

$\displaystyle U_{m}=\frac{r_{m}}{\beta_{m}-\alpha_{m}W_{m-1}}U_{m+1}+\frac{b_{m}+\alpha_{m} g_{m-1}}{\beta_{m}-\alpha_{m} W_{m-1}}$

is obtained, and so

$\displaystyle W_{m}=\frac{r_{m}}{\beta_{m}-\alpha_{m}W_{m-1}},g_{m}=\frac{b_{m}+\alpha_{m}g_{m-1}}{\beta_{m}-\alpha_{m}W_{m-1}}$

If $ U_{0}=0$, then $ W_{0}=g_{0}=0$, in order that the difference relation

$\displaystyle U_{0}=W_{0}U_{1}+g_{0}$

holds for any $ U_{1}$. The remaining $ W_{m}, g_{m}$ can now be computed as

$\displaystyle W_{1}=\frac{r_{1}}{\beta_{1}}\quad \qquad g_{1}=\frac{b_{1}}{\beta_{1}}$

$\displaystyle W_{2}=\frac{r_{2}}{\beta_{2}-\alpha_{2}W_{1}}\quad \qquad g_{2}=\frac{b_{2}+\alpha_{2}g_{1}}{\beta_{2}-\alpha_{2}W_{1}}$

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If $ U_{M}=Y,$ then $ U_{1},U_{2},......., U_{M-1}$ are computed as

$\displaystyle U_{M-1}=W_{M-1} Y + g_{M-1}$

$\displaystyle U_{M-2}=W_{M-2}U_{M-1}+g_{M-2}$

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$\displaystyle U_{1}=W_{1}U_{2} + g_{1}$

In using this method, substantial errors will appear in the computed values of$ U_{1},U_{2},........,U_{M-1}$ unless

Now

$\displaystyle W_{1}=\frac{r_{1}}{\beta_{1}}\leq 1$

$\displaystyle W_{2}=\frac{r_{2}}{\beta_{2}-\alpha_{2}W_{1}}\leq \frac{r_{2}}{\beta_{2}-\alpha_{2}}\leq 1$

and so on, since $ \alpha_{ m}>0,\quad \beta_{m}>0,\quad r_{m}>0,$ $ \beta_{m}\geq(\alpha_{m}+r_{m}) \quad for \quad 1\leq m\leq M-1$. This leads to $ 0<W_{m}\leq 1 \quad (1\leq m\leq M-1)$.

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