Solving Parabolic Equations in 2 and 3 Space Dimensions

Resumé

In ‘Numerical Solution of Parabolic Equations’ [18] we have introduced the general theory for finite difference methods for parabolic problems. Inspired by a class of specific problems we shall now extend the theory to address the challenges of some realistic problems. We discuss a number of extensions of the ADI methods to three space dimensions and compare their various merits w.r.t. accuracy and stability in addition to their tendency to produce unwanted oscillations and spikes. Our method of choice is an extension of the SDR-method which will produce smooth solutions. SDR is only first order in time but in the presence of a singularity the meaning of order loses some of its significance. We show that by using extrapolation techniques it is possible to recover and in fact raise the order without losing the smoothness. In order to treat boundary layer problems we use variable steps in space: the exponentially expanding steps. We analyze the stability and accuracy of exponentially expanding steps and show that the global error (in space) is proportional to ( − 1)2 where is the expansion factor. With this information we can again use extrapolation
techniques to produce accurate solutions with a modest time consumption.