Suppression of Oscillations in Godunov’s Method for a Resonant Non-Strictly Hyperbolic System

Abstract

We prove the stability of the 2 x 2 Godunov numerical method in a resonant nonlinear system of conservation laws. The system we study provides one of the simplest settings in which wave speeds can coincide in a nonlinear problem, namely, an inhomogeneous equation in which the inhomogeneity is treated as an unknown variable: ut + f(a,u)x O, at O. This is a model for resonance in more complicated systems, such as transonic flow in a variable area duct, and certain resonant problems in multiphase flow and elasticity. We show that the total variation of the conserved quantities can grow at most linearly in weak solutions generated by the 2 2 Godunov method, under the assumption that a(x) satisfies the threshold smoothness condition that the total variation of a(x) is finite. We show by counterexample that the condition is sharp in the sense that there is no bound on the growth rate based on the C norm of this variable. This is the most complicated setting in which the stability of the 2 2 Godunov method has been demonstrated, and our results provide the first such result for a numerical method that is (essentially) based on the solution of the Riemann problem for a resonant non-strictly hyperbolic system. The solution of the Riemann problem is more interesting when wave speeds coincide because the coordinate system of Riemann invariants is singular, and there exists a multiplicity of possible time asymptotic wave patterns for solutions. As a consequence, numerical methods that are based on the solution of the Riemann problem can introduce spurious oscillations in the approximate solutions. Indeed, counterexamples show that the total variation in u of the waves in the Riemann problem step of the 2 x 2 Godunov method can tend to infinity as Ax 0, even when the initial total variation in u is zero. Thus our results verify that the averaging step in the Godunov method wipes out the numerical oscillations that can occur in the Riemann problem solution step of the method. We interpret this as demonstrating that numerical methods based on the Riemann problem are viable in this non-strictly hyperbolic setting.