8.3 NUMERICAL SOLUTION TECHNIQUES

Many techniques have been developed for the numerical solution of partial differential equations. The best method depends on the type of PDE being solved and on the geometry of the system. Partial differential equations having the form of Equation (8.12) are known as parabolic PDEs and are among the easiest to solve. We give here the simplest possible method of solution, one that is directly analogous to the marching-ahead technique used for ordinary differential equations. Other techniques should be considered (but may not be much better) if the computing cost becomes significant. The method we shall use is based on finite difference approximations for the partial derivatives. Finite element methods will occasionally give better performance, although typically not for parabolic PDEs.

The technique used here is a variant of the method of lines in which a PDE is converted into a set of simultaneous ODEs. The ODEs have z as the independent variable and are solved by conventional means. We will solve them using Euler s method, which converges O( ? z). Higher orders of convergence, e.g., Runge-Kutta, buy little for reasons explained in Section 8.3.3. The ODEs obtained using the method of lines are very stiff, and computational efficiency can be gained by using an ODE-solver designed for stiff equations. However, for a solution done only once, programming ease is usually more important than computational efficiency.