Construction of the limiter based on averaging of solutions for discontinued Galerkin method

The Galerkin method with discontinuous basis functions proved to be well established in the numerical solution of hyperbolic systems of equations. However, to ensure the monotonicity of the solution obtained by this method, it is necessary to use a smoothing operator, especially if the solution contains strong discontinuities. In this paper we explore a well-proven smoothing operator based on WENO reconstruction and a smoothing operator of a new type based on averaging of the solution, taking into account the rate of change of the solution and the rate of change of its derivatives. Comparison of the effect of these limiters in solving a series of test problems is presented. It is shown that the application of the proposed smoothing operator is not inferior to the action of the WENO limiter, and in some cases exceeds the accuracy of the received solution, which is confirmed by numerical studies.