2.5.1 Thermal Conduction in 1D

Again, discretize the equation of thermal conduction,

$$\frac{\partial T(x,t)}{\partial t} = \lambda \frac{\partial^{2} T(x,t)}{\partial x^{2}}$$

Earlier we applied DNGF to the l.h.s. and DDST at time $t_{n}$ to the r.h.s.:

$$\frac{\partial T(x,t)}{\partial x^{2}} \approx \frac{\delta_{i}^{2}T_{i}^{n}}{(\Delta x)^{2}}$$

In this manner we arrived at the ``FTCS-''formula.

Now we may use the DDST formula at time $t_{n+1}$,

$$\frac{\partial T(x,t)}{\partial x^{2}} \approx \frac{\delta_{i}^{2}T_{i}^{n+1}}{(\Delta x)^{2}}$$

This leads us to the ``implicit scheme of first order''

$$\frac{1}{\Delta t} [T_{i}^{n+1}-T_{i}^{n}] =\frac{\lambda}{(\Delta x)^{2}} [T_{i+1}^{n+1}-2T_{i}^{n+1}+T_{i-1}^{n+1}]$$

which may be written, using $a \equiv \lambda\,\Delta t/(\Delta x)^{2}$,

$$-a T_{i-1}^{n+1}+(1+2a)T_{i}^{n+1}-aT_{i+1}^{n+1}=T_{i}^{n}$$

or

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

where (for fixed $T_{0}$ and $T_{N}$)

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

Invert this tridiagonal system by the Recursion Method.



EXERCISE: Redo the earlier exercise on One-dimensional thermal conduction by applying the implicit scheme in place of the FTCS method. Use various values of $\Delta t$ (and therefore $a$.) Compare the efficiencies and stabilities of the two methods.