Error bounds for finite-difference approximations for certain nonlinear parabolic systems in the quarter-plane.

Abstract

This paper is a continuation of the work of Joel Smoller, Takaaki Nishida, and David Hoff in analyzing certain nonlinear parabolic systems with initial data of bounded variation via the method of finite-difference approximation. Whereas they considered for their domain the half-plane in space-time, we make the natural progression to the quarter-plane. We also outline the basic facts needed for the next logical case, where the domain for the space variable is a finite interval. Using the classical technique of Oleinik, our finite-difference approximations are shown to converge lo the unique solution of the system as the mesh parameters approach zero. This mirrors the work of Nishida and Smoller. Then, as in the work of Hoff and Smoller, we establish error bounds for the rate of that convergence. Since the systems under consideration are nonlinear inhomogeneous versions of the heat equation, of primary importance is our analysis of the discrete version of the heat equation. The properties of its fundamental solution closely correspond to those of the familiar one-dimensional heat kernel, and are important tools in our work. In an addendum, we improve upon the basic heat equation error estimate of Hoff and Smoller, thereby sharpening their error estimate for the nonlinear system in the half-plane, as well as ours for the system in the quarter-plane.