A partial upwind difference scheme for nonlinear parabolic equations

摘要

In this paper, we present an explicit partial upwind difference scheme for the problem: ut − duxx + f(u))x + g(x, u) = 0, d > 0, 0 ⩽ x ⩽ 1, t ⩾ 0, with u(x, 0), u(0, t) and u(1, t) all prescribed. Formulas for the upwind factors are provided. We use the method of lower and upper solutions and the theory of M-matrices. The present method is more accurate and allows larger time steps than the corresponding one-sided method. We obtain the order of convergence for our scheme. We also prove the convergence of the time evolving solutions to the unique steady-state solution as the time approaches infinity. The method can be generalized to multi-dimensional analogues. To support the theory, numerical results are given.