Limiting Functions Affecting the Accuracy of Numerical Solution Obtained by Discontinuous Galerkin Method

Abstract

In the numerical solution of hyperbolic systems of equations, Galerkin method with discontinuous basic functions is proved to be very reliable. However, to ensure the monotony of the solution obtained by this method, it is necessary to use a smoothing operator, especially if the solution contains strong discontinuities. In this chapter, we consider the classic Cockburn limiter, a moment limiter that preserves the high order of the scheme, well-proven smoothing operator based on Weighted Essentially Non-Oscillatory (WENO) reconstruction, the smoothing operator of a new type based on averaging solutions, taking into account the rate of change of the solution and the rate of change of its derivatives and slope limiter, preserving the positivity of pressure. A comparison was made of the actions of these limiters on a series of test problems. Numerical results show that using discontinuous Galerkin method and applying moment limiter, slope limiter, WENO limiter, or limiter based on averaging allows to obtain a high order of accuracy on smooth solutions, as well as the clear, non-oscillating profiles on shock waves provided with appropriate constants for the correct determinations of defective cells. In addition, slope limiter, WENO limiter, and averaging limiter are simple enough to implement and be generalized on the multidimensional unstructured grids.