Central Schemes: A Powerful Black-Box Solver for Nonlinear Hyperbolic PDEs

Abstract We review a class of Godunov-type finite-volume methods for hyperbolic systems of conservation and balance laws—nonoscillatory central schemes. These schemes date back to 1950s, when the first-order Lax–Friedrichs scheme was introduced. The central Lax–Friedrichs scheme can be viewed as a simple alternative to the upwind Godunov scheme, which was also introduced in the 1950s. The main idea in the construction of both central and upwind first-order schemes is the same: use a global piecewise constant approximation of the solution at a certain time level and evolve it in time to the next time level exactly. The exact evolution is performed using the integral form of the studied system of PDEs. The difference is in one small detail—which is in fact not small at all—how to select the space–time control volume for the time evolution. The key idea in the construction of central schemes is to choose these control volume in such a way that no (localized) Riemann problems need to be solved at the evolution step. This makes central schemes particularly simple and universal numerical tool for general hyperbolic systems. On the other hand, central schemes are based on averaging the nonlinear waves rather than resolving them and thus they have larger numerical dissipation than their upwind counterparts. In order to increase the resolution achieved by central schemes, one has to increase their order. We describe how to design high-order nonoscillatory central schemes and also discuss how to further decrease their numerical dissipation without risking oscillations. The latter is achieved by utilizing some upwinding information (local speeds of propagation) within the framework of the Riemann-problem-solver-free central schemes and modifying the set of control volumes used for the time evolution. This leads to another type of central schemes—central-upwind schemes, whose derivation is reviewed in this work.