Computing interface Motion in Compressible Gas Dynamics

A “Hamilton-Jacobi” level set formulation of the equations of motion for propagating interfaces has been introduced recently by Osher and Sethian. This formulation allows fronts to self-intersect, develop singularities, and change topology. The numerical algorithms based on this approach handle topological merging and breaking naturally, work in any number of space dimensions, and do not require that the moving front be written as a function. Instead, the moving front is embedded as a particular level set in a fixed domain partial differential equation. Numerical techniques borrowed from hyperbolic conservation laws are then used to accurately capture the complicated surface motion that satisfies the global entropy condition for propagating fronts given by Sethian. In this paper, we analyze the coupling of this level set formulation to a system of conservation laws for compressible gas dynamics. We study both conservative and non-conservative differencing of the level set function and compare the two approaches. To illustrate the capability of the method, we study the compressible Rayleigh-Taylor and Kelvin-Helmholtz instabilities for air-air and air-helium boundaries. We perform numerical convergence studies of the method over a range of parameters and analyze the accuracy of this approach applied to these problems.