Towards the ultimate conservative difference scheme I. The quest of monotonicity

Abstract

Many interesting flows, notably those containing shocks, can be computed with conservative, dissipative difference schemes based on Eq. (i). In order to analyze or design such schemes it is most practical to start from the single convection equation ~w ~w ~ + a ~x = O. (2) How to make a scheme for Eq. (2) useful to integrate Eq. (1) is explained by Van Leer [4]. The best-known conservative schemes, those of Lax, Godunov and Lax-Wendroff, are based on the cluster of nodal points C1 defined in Fig. 1. t o CI o o o o X I X 0 Xj Figure 1. The cluster C1 C2 o o o o o o o o o o L L I [ L L ] I I I x, x~x o x~ x, x.z x., x x o x, o o o b i X 0 X I °c I x 2 Figure 2. Some clusters suitable for conservative monotonic schemes o o oC4[ Eq. (2) transforms an initially monotonic distribution such that it remains monotonic at later times; in fact, the original distribution is just shifted over a distance axt. A difference scheme for Eq. (2) can not deliver this exact solution, except for integer values of ~ = aAt/Ax, but it may at least be required to produce monotonic results for all stable values of (~. This requirement is called the monotouicity condition; schemes satisfying it will colloquially be called monotonic schemes.