Chapter 6 - Higher-order schemes

Abstract

This chapter discusses the ability of the original Godunov scheme to capture shocks and discontinuities in the computed variables. The low order of the scheme induces a dramatic loss of accuracy when the Courant number becomes small. Computational efficiency can be restored only by using higher-order schemes. The only way of suppressing the artificial oscillations that usually appear near steep fronts is to apply a non-standard treatment to the cells where the gradient of the variables is large. One possibility to apply a smoothing to a non-monotonic solution computed by the higher-order scheme, is in the pseudo-viscosity method. Two types of techniques can be distinguished—flux-limiting techniques that prevent oscillations by limiting the fluxes, and slope limiting techniques that limit the variations of the flow variables. Godunov-type algorithms count six steps—discretization of the cells in finite volumes, profile reconstruction over the cells, determination of the equivalent Riemann problems at the cell interfaces, solution of the Riemann problems, computation of the fluxes at the cell interfaces, and flux balance to determine the variables at the next time step. Higher-order schemes differ from the original Godunov scheme only in the reconstruction of the profile and in the determination of the equivalent Riemann problems.