A partial upwind difference scheme for nonlinear parabolic equations
In many diffusion processes arising in physical problems, essentially the transport term plays the dominant role. In these cases, it is known that the centered differencing of the first order term in space variable in the governing nonlinear parabolic transport equation gives rise to oscillatory solutions at reasonable grid sizes. One-sided or ``upwind'' schemes can give oscillation free solutions. However, in general, full ``upwind'' schemes introduce large amounts of artificial diffusions in contrast to the centered schemes. This paper presents a weighted difference scheme which is a generalization of the scheme due to \textit{S. Osher} [SIAM J. Numer. Anal. 18, 129-144 (1981; Zbl 0471.65069)]. Here two parameters \(a_ 1\), \(a_ 2\) in the difference approximation of the nonlinear first order term are used. \((a_ i=1\) leads to Osher's full upwind form, and \(a_ i=0\) corresponds to the centered form.) Using the method of lower and upper solutions and the theory of M-matrices, the scheme is found to be first- order accurate in the space variable. If Osher's full upwind scheme is used, only first-order approximation is found, too. But the actual errors are smaller for partial upwind difference schemes. To support this, a numerical example for a Burger's equation is given.
scientific article; zbMATH DE number 4108083