A multidimensional generalization of Roe's flux difference splitter for the euler equations

Abstract

Upwind methods for the 1-D Euler equations, such as TVD schemes based on Roe's approximate Riemann solver, are reinterpreted as residual distribution schemes, assuming continuous piecewise linear space variation of the unknowns defined at the cell vertices. From this analysis three distinct steps are identified, each allowing for a multidimensional generalization without reference to dimensional splitting or 1-D Riemann problems. A key element is the necessity to have continuous piecewise linear variation of the unknowns, requiring linear triangles in two space dimensions and tetrahedra in three space dimensions. Flux differences naturally generalize to flux contour integrals over the triangles. Roe's flux difference splitter naturally generalizes to a multidimensional flux balance splitter if one assumes that the parameter vector variable is the primary dependent unknown having linear variation in space. Nonlinear positive and second-order scalar distribution schemes provide a true generalization of the TVD schemes in one space dimension. Although refinements and improvements are still possible for all these elements, computational examples show that these generalizations present a new framework for solving the multidimensional Euler equations.