Constructing a Limiter Based on Averaging the Solutions for the Discontinuous Galerkin Method

Abstract

The Galerkin method with discontinuous basis functions has proved to be effect in solving hyperbolic systems of equations numerically. However, to ensure the solution yielded by this method is monotonic, a smoothing operator is required to be used, especially if the solution contains strong discontinuities. In this paper, a well-proven smoothing operator based on a WENO reconstruction and a smoothing operator of a new type based on averaging the solutions that takes into account the rate of change of the solution and the rate of change of its derivatives is considered. The effect of these limiters in solving a series of test problems is compared. The application of the proposed smoothing operator is shown to be as good as the action of a WENO limiter, in some cases even exceeding it in the accuracy of the resulting solution, which is confirmed by the numerical studies.