Numerical Solution

Various numerical methods have been devived to solve the heat equation numerically. The method based on Finite Difference start with rewriting the heat equation in terms of finite differences on a space grid (equally spaced, $\Delta_x$) and a time grid (equally spaced, $\Delta_t$).

Picture a $1-D$ space and time grid as in the following figure;

The initial temperature field $T(s,t=0)$ is on the $t = 0$ line. It must satisfy the boundary conditions. The task is now to solve for $T(x,t)$ for all the subsequent time lines under the BCs constraints.

Note that the grid does not close with a known field profile for large time. This makes it impossible to use an algorithm based on a stencil (averaging) as in the elliptic PDE scheme.

An explicit scheme follows if we use a first order time derivative, i.e., the forward derivative,

$$\frac{\partial T(x, t)}{\partial t} = \frac{T(x,t+\Delta t) - T(x,t)}{\Delta t}$$ (3)

The space derivative can make use of the finite difference second order form. Therefore, the heat equation becomes

$$\frac{T(x,t+\Delta t) - T(x,t)}{\Delta t} = \frac{k}{C \r... ...x+\Delta_x), t) - 2 T(i,j) + T(x-\Delta_x, t) } { \Delta_x^2 }$$ (4)

Introducing the grid indices, $x = i \Delta_x$ and $t = j \Delta_t$, the heat equation becomes

$$T_{i, j+1} = T_{i, j} + \kappa \left[ T_{i+1.j} - 2 T_{i, j} + T_{i-1, j} \right]$$ (5)

where $\kappa = \frac{k \Delta t}{c \rho {\Delta t}^2 }$.

This algorithm is explicit. The $j$ index starts at $t = 0$, the initial time, with $T_{ i, j=0}$ , and proceeds up to the desired time.

This yields the finite difference equation for heat transfer.