One-Dimensional Front Tracking Based on High Resolution Wave Propagation Methods

We present a fully conservative, high resolution approach to front tracking for nonlinear systems of conservation laws in two space dimensions. An underlying uniform Cartesian grid is used, with some cells cut by the front into two subcells. The front is moved by solving a Riemann problem normal to each segment of the front and using the motion of the strongest wave to give an approximate location of the front at the end of the time step. A high resolution nite volume method is then applied on the resulting slightly-irregular gird to update all cell values. A \large time step" wave propagation algorithm is used that remains stable in the small cut cells with a time step that is chosen with respect to the uniform grid cells. Numerical results on a radially symmetric problem show that pointwise convergence with order between 1 and 2 is obtained in both the cell values and location of the front. Other computations are also presented. 1. Introduction. We will describe a fairly simple approach to front tracking in two space dimensions, giving a general formulation and then concentrating on shock tracking for the Euler equations of gas dynamics. The method is fully conservative and based on modern high-resolution shock capturing methods. An underlying uniform Cartesian grid is used, with some rectangles subdivided into two or more computational cells where discontinuities in the solution are expected (see Figure 9 for an example grid). A high resolution nite volume method is applied on the resulting grid, based on the solution of Riemann problems and appropriate slope limiters. This method is implemented in a \wave-propagation" form, as developed in 31, 35], for example. The cell average at the end of a time step is computed from the cell average at the beginning of the step with modiications due to all waves that enter the cell. The waves come from solving one-dimensional Riemann problems at each cell interface in the direction normal to the interface. Transverse propagation of these waves is also introduced to improve numerical stability and incorporate the cross-derivative terms needed for second order accuracy. An advantage of the wave propagation form is that reasonable time steps can be taken even if some of the subcells created by the interface are orders of magnitude smaller than the uniform Cartesian cells. Uniform time steps are used throughout the computation, with the time step chosen so that the …