Fast fourier transform method for partial differential equations, case study: The 2-D diffusion equation

AbstractThe solution of time-dependent partial differential equations using discrete (i.e. finite difference) versions of those equations traditionally has been done by marching forward in time, one time unit per step. The authors here extend earlier results to higher dimensions, focusing on the solution of the discrete diffusion or “heat” equation in 2-D, and show that the solution may be found using fast Fourier transform (FFT) techniques that treat the solution as the output of a linear filter. This technique is analogous to an integral method for the differential equation, and is related to the Green's function of the difference equation. It allows the solution to be determined with a “single-step” computation that does not require the solution at intermediate time levels. The number of numerical operations is essentially independent of the time level at which the solution is desired. The results of numerical examples are given and computation times are shown to be much faster than those of the traditional method for all but small time levels