Multidimensional upwinding. Part I. The method of transport for solving the Euler equations

The aim of this paper is to show a new approach towards the discretization of multidimensional conservation laws. The idea of transport associated with the solution of a scalar equation is used for the convective part of the compressible Euler equations. A multidimensional wave structure is derived to model the acoustic part of this non-linear system, that allows infinitely many propagation directions in the numerical method. This provides the basic knowledge to construct a numerical method that does not rely on Riemann solvers. A more general definition of the waves, together with the concept of consistency. enables the design of a number of effective, genuinely multidimensional, methods. (C) 1998 Academic Press.