Selected Topics in Approximate Solutions of Nonlinear Conservation Laws. High-Resolution Central Schemes

Abstract

Central schemes offer a simple and versatile approach for computing approximate solutions of non-linear systems of hyperbolic conservation laws and related PDEs. The solution of such problems often involves the spontaneous evolution of steep gradients. The multiscale aspect of these gradients poses a main computational challenge for their numerical solution. Central schemes utilize a minimal amount of information on the propagation speeds associated with the problems, in order to accurately detect these steep gradients. This information is then coupled with high–order, non-oscillatory reconstruction of the approximate solution in ‘the direction of smoothness’: that is, information of smoothness does not cross regions of steep gradients. The use of central stencils enables us to realize the reconstructed solutions through simple quadratures. In this manner, central schemes avoid the intricate and time-consuming details of the eigen-structure of the underlying PDEs, and in particular, the use of (approximate) Riemann solvers, dimensional splitting, etc. The resulting family of central schemes offers relatively simple, black-box solvers for a wide variety of problems governed by multi-dimensional systems of non-linear hyperbolic conservation laws and related convection-diffusion problems.