5.2.2 Implicit Scheme of First Order

Take the second spatial derivative at time $t_{n+1}$ instead of $t_{n}$:

$$\frac{1}{\Delta t} \left[ u_{j}^{n+1}-u_{j}^{n}\right] = \frac{\lambda}{(\Delta x)^{2}} \left[ u_{j+1}^{n+1}-2 u_{j}^{n+1}+u_{j-1}^{n+1}\right]$$

Again defining $a \equiv \lambda \Delta t / (\Delta x)^{2}$, we find, for each space point $x_{j}$ $\;(j=1,2,..N-1)$,

$$\fbox{$ \displaystyle -a u_{j-1}^{n+1}+(1+2a)u_{j}^{n+1}-a u_{j+1}^{n+1}=u_{j}^{n} $}$$

Let the boundary values $u_{0}$ and $u_{N}$ be given; the set of equations may then be written as

$$\mbox{${\bf A}$} \cdot \mbox{$\bf u$}^{n+1} = \mbox{$\bf u$}^{n}$$

with

$$\mbox{${\bf A}$} \equiv \left( \begin{array}{cccccc} 1 & 0 & 0 & . & . & 0 \\ -a & 1+2a ... ...& . \\ . & . & . & -a & 1+2a & -a \\ . & . & . & 0 & 0 & 1 \end{array}\right)$$

$\Longrightarrow$Solve by Recursion!

Stability:

We find

$$g = \frac{1}{1+4a\,\sin^{2}(k \Delta x/2)}$$

Since $\vert g\vert \leq 1$ under all circumstances, we have here an unconditionally stable algorithm!





EXERCISE: Apply the implicit technique to the thermal conduction problem discussed before. Consider the efficiency of the procedure as compared to FTCS. Relate the problem to the Wiener-Levy random walk.