A Problem-Independent Limiter for High-Order Runge–Kutta Discontinuous Galerkin Methods

This paper is devoted to the use of discontinuous Galerkin methods to solve hyperbolic conservation laws. The emphasis is laid on the elaboration of slope limiters to enforce nonlinear stability for shock-capturing. The objectives are to derive problem-independent methods that maintain high-order of accuracy in regions where the solution is smooth, and in the neighborhood of shock waves. The aim is also to define a way of taking into account high-order space discretization in limiting process, to make use of all the expansion terms of the approximate solution. A new slope limiter is first presented for one-dimensional problems and any order of approximation. Next, it is extended to bidimensional problems, for unstructured triangular meshes. The new method is totally free of problem-dependence. Numerical experiments show its capacity to preserve the accuracy of discontinuous Galerkin method in smooth regions, and to capture strong shocks.