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A numerical technique, composed of the Van Leer-Tai-Powell optimization and a modified procedure, is applied to design multistage schemes that give optimal damping of high frequencies for given upwind-biased spatial differencing with implicit and explicit residual smoothing. The analysis is done for a scalar convection equation in one space dimension. The object of this technique is to make the schemes suited for multigrid acceleration. The optimal multistage schemes and their damping properties are presented in this paper. By keeping the multistage coefficients from the one-dimensional analysis and simply redefining the Courant number, the schemes can be applied to multidimensional problems. A fast Euler code is used to evaluate the suitability of the schemes for multidimensional multigrid computations. Numerical results show that the modified schemes effectively enhance the convergence performance on both single and multiple grids.

Optimal multistage schemes for Euler equations with residual smoothing