Diffusion Equation

• The first field problem we're going to tackle will be very simple: distribution of temperature on a square hot plate with some edges of the plate kept hot at some fixed temperature, and other edges kept cold, also at some fixed temperature.
• We'll seek a static solution, i.e., our PDE is going to be:
  
  $$\nabla^2 T(x, y) = \frac{\partial^2 T(x,y)}{\partial x^2} + \frac{\partial^2 T(x,y)}{\partial y^2} = 0$$
  
  

• As you recall from lectures on numerical analysis, that you have undoubtedly been attending, the meaning of a Laplace operator is that at every point (x₀, y₀) it averages T(x,y) over the neighbourhood of that point.
• Discretised form of $\nabla^2 T(x, y) = 0$:
  
  $$\begin{eqnarray*}\nabla^2 T &\equiv& \frac{\partial}{\partial x} \frac{T(x_0 +... ..., y_0 + h) + T(x_0, y_0 - h) - 4 T(x_0, y_0) \big) \\ &=& 0 \end{eqnarray*}$$
  
  This equation has the following simple solution:
  

  $$\frac{1}{4} \big( T(x_0 + h, y_0) + T(x_0 - h, y_0)$$ T(x₀, y₀) =

  $$\qquad + T(x_0, y_0 + h) + T(x_0, y_0 - h) \big)$$ (3.19)

  
  

• This solution is the essence   of the Jacobi iteration method that is always quoted in this context and scorned upon. But there is nothing wrong with it. It is slow but simple and very stable, and it provides an ideal demonstration for High Performance Fortran and for Fortran-90.