Chapter 8 - Large-time-step algorithms

Abstract

This chapter provides an overview of the principles of front tracking algorithms describing a classical explicit/implicit scheme. The timeline interpolation method with computational examples is also explained. The chapter then discusses the treatment of the inner cells, treatment of boundary conditions, treatment of shocks, treatment of rarefaction waves, treatment of sonic points, and principle of the method to hyperbolic systems of conservation laws. Courant number is small throughout the simulation because the time step is constrained by the faster waves. Explicit schemes with variable stencils can be divided into two categories—flux based characteristics schemes for linear advection and front tracking schemes for non-linear conservation laws. In these methods, the average flux over the time step at the interface between two computational cells is computed by averaging the variable over the whole domain of dependence of the interface. The domain of dependence is determined by tracing back the characteristic lines across several computational cells. Characteristic backtracking may become an ill-posed problem in the presence of shock, and the front tracking techniques follow the various waves arising from the Riemann problems at the cell interfaces forward in time. As the number of Riemann problems is finite, the number of waves is also finite and the solution can be computed for infinite times in a finite number of steps.