Abstract
In this paper, we present an explicit partial upwind difference scheme for the problem: u t − du xx + 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.
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