Mesh refinements for parabolic equations of second order

Abstract:

Given certain implicit difference approximations to ${u_t} = a(x){u_{xx}} + b(x){u_x} + c(x)u$ in the region $- \infty < x < \infty ,t \geqq 0$, with a finer x-mesh width in the left half-plane than in the right, we consider the stability in the maximum norm of these schemes using several different interface conditions (at $x = 0$). In order to obtain our results, we first prove a stability theorem for certain simple second-order parabolic initial boundary systems on an evenly spaced mesh in the right half-plane alone. By a standard procedure, the first problem is converted into the second one, and solved in this manner.