An Unconditionally Stable Method for the Euler Equations

We discuss how to combine a front tracking method with dimensional splitting to solve numerically systems of conservation laws in two space dimensions. In addition we present an adaptive grid reenement strategy. The method is unconditionally stable and allows for moderately high cfl numbers (typically 1{4), and thus it is highly eecient. The method is applied to the Euler equations of gas dynamics. In particular, it is tested on an expanding circular gas front, a wind tunnel with a step, a double Mach reeection as well as a shock-bubble interaction. The method shows very sharp resolution of shocks. 1. Introduction Front tracking has proved to be an eecient tool to analyze rigorously hyperbolic conservation laws, both scalar equations and systems, in one space dimension. We here show how to use front tracking for systems of conservation laws as an unconditionally stable numerical method in two space dimensions, and we test it on the Euler equations of gas dynamics. Let us rst discuss the method of front tracking. Consider the hyperbolic conservation law u t + f(u) x = 0; uj t=0 = u 0 : (1) If we approximate the initial data by a step function, that is, a piecewise constant function, the problem is locally reduced to solving a series of problems with Riemann initial data, i.e., a single jump separating two constant states. In the case of general systems, the solution of a Riemann problem is locally characterized by the Lax construction where one has a set of wave curves forming a local system of coordinates around two nearby constant states. Shocks and contact discontinuities are unchanged in the front tracking technique. However, if two states are connected by a rarefaction wave, we sample points along the rarefaction curve and approximate the (continuous) rarefaction wave by a (discontinuous) function with several small jumps. In this way the approximate solution is a step function for any xed time. We denote all discontinuities in the solution as fronts. When two fronts collide we will have another local Riemann problem, which can