CHAPTER 19 - PARABOLIC EQUATIONS IN TWO DIMENSIONS: I

Abstract

This chapter discusses parabolic equations in two dimensions, and discusses compatibility, stability, and convergence. Parabolic equations have their main physical origin in the problems of diffusion and heat conduction. Stability can be examined either by the Fourier method or by examining the latent roots of relevant matrices. With partial differential equations of initial-value type, there is a phenomenon that has no counterpart in ordinary differential equations, in that successive refinement of the interval length can give a finite-difference solution that is stable but can converge to the solution of a different differential equation. The finite-difference equation is compatible with the differential equation if the local truncation errors tend to zero. Compatibility and stability usually imply convergence. The solution of the partial differential equation involves a combination of decreasing exponentials.