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An accurate finite difference scheme for solving convection-dominated diffusion equations. (English) Zbl 0892.76050

Summary: Approximating convection-dominated diffusion equations requires a very accurate scheme for the convection term. The most famous is the method of backward characteristics, which is very precise when a good interpolation procedure is used. However, this method is difficult to implement in two or three dimensions. The goal of this paper is to show that it is possible to construct finite difference schemes almost as accurate as the method of characteristics. Starting from a family of second- and third-order Lax-Wendroff-type schemes, a TVD and \(L^\infty\)-stable scheme that is easy to implement in higher dimensions is constructed. Numerical tests are performed on various model problems whose solutions are known and on classical problems. Comparisons with some other limiter schemes and the method of characteristics are discussed.

MSC:

76M20 Finite difference methods applied to problems in fluid mechanics
76R50 Diffusion
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