Numerical transport of an arbitrary number of components

This paper deals with the numerical transport of an arbitrary number of materials having the same velocity. One difficulty is to derive numerical algorithms that are conservative for the mass of each component and that satisfy some inequality and equality constraints: each mass fraction has to stay in [0, 1] and the sum of all mass fractions should be 1. These constraints are satisfied by the classical upwind scheme (which is very dissipative) but not for most of nonlinear (high-order or anti-dissipative) schemes. Here we propose local conditions of inequality type for the finite volume fluxes of mass fractions to ensure the aforementioned constraints. More precisely, we give explicit stability intervals for each flux. This is done in the manner of Després and Lagoutière [B. Després, F. Lagoutière, Contact discontinuity capturing schemes for linear advection and compressible gas dynamics, J. Sci. Comput. 16 (4) (2002) 479–524] for hyperbolic systems, Després and Lagoutière [B. Després, F. Lagoutière, Numerical resolution of a two-component compressible fluid model with interfaces, Progress Comput. Fluid Dyn., to appear] for the transport of 2 components; see also Bouchut [F. Bouchut, An anti-diffusive entropy scheme for monotone scalar conservation laws, J. Sci. Comput. 21 (1) (2004) 1–30] for the same type of inequality constraints for nonlinear conservation laws. Comparisons on two dimensional test-cases with the Youngs’ interface reconstruction algorithm [D.L. Youngs, An interface tracking method for a 3D eulerian hydrodynamics code, Technical Report 44/92/35, AWRE, 1984] show that results are qualitatively comparable. The advantages of this approach are its simplicity, its low computational cost, and its flexibility since it can deal with interfaces as well as mixing zones.