Toward Front Tracking Based on Conservation in Two Space Dimensions

Abstract

A two-dimensional front tracking method based on conservation for scalar conservation law and for the Euler system of gas dynamics is being developed. In the method, discontinuities are tracked by enforcing the conservation properties of the PDEs. Unlike the traditional front tracking methods, for which the conservation is only a property that may or may not be preserved, in this front tracking method the conservation is the mechanism by which the tracking is performed. The method is also extended to treat the reflection wall boundary in the Euler system. As an attempt to treat the interactions of discontinuities by enforcing the conservation properties without solving two-dimensional Riemann problems, three special cases of treatment of discontinuity interactions in scalar conservation law and in the Euler system of gas dynamics are studied. Finally, numerical experiments are implemented in both scalar and system cases to show the efficiency of the method.