Difference Methods for PDE in Several Dimensions

Abstract

Finite-difference methods for partial differential equations in several dimensions are presented by first handling the basic (parabolic) diffusion equation in two dimensions by explicit and implicit difference schemes, allowing us to introduce the corresponding stability criteria. The treatment of alternating direction implicit (ADI) schemes is supplemented by the extension to three space dimensions. Several schemes for solutions of hyperbolic equations are listed. Solving elliptic equations (Poisson and Laplace) by classical relaxation methods (Jacobi, Gauss–Seidel, SOR, SSOR) is discussed, as well as high-resolution schemes for hyperbolic equations. The need to respect the underlying nature of the physical problems is emphasized by solving the two-dimensional diffusion and Poisson equations in polar coordinates with various boundary conditions. The boundary element method, the finite element method, and a mesh-free method based on radial basis functions are presented. In addition to many variants of Laplace and Poisson equations, the Problems include several instances of the linear diffusion equation, as well as a non-linear case describing the growth of bacterial biofilms.