Simplified Discretization of Systems of Hyperbolic Conservation Laws Containing Advection Equations

Abstract

The high speed flow of complex materials can often be modeled by the compressible Euler equations coupled to (possibly many) additional advection equations. Traditionally, good computational results have been obtained by writing these systems in fully conservative form and applying the general methodology of shock-capturing schemes for systems of hyperbolic conservation laws. In this paper, we show how to obtain the benefits of these schemes without the usual complexity of full characteristic decomposition or the restrictions imposed by fully conservative differencing. Instead, under certain conditions defined in Section 2, the additional advection equations can be discretized individually with a nonconservative scheme while the remaining system is discretized using a fully conservative approach, perhaps based on a characteristic field decomposition. A simple extension of the Lax–Wendroff Theorem is presented to show that under certain verifiable hypothesis, our nonconservative schemes converge to weak solutions of the fully conservative system. Then this new technique is applied to systems of equations from compressible multiphase flow, chemically reacting flow, and explosive materials modeling. In the last instance, the flexibility introduced by this approach is exploited to change a weakly hyperbolic system into an equivalent strictly hyperbolic system, and to remove certain nonphysical modeling assumptions.