A filtering monotonization approach for DG discretizations of hyperbolic problems

TL;DR

This paper presents a filtering technique for Discontinuous Galerkin methods that reduces spurious oscillations near discontinuities in hyperbolic problems by adaptively switching between high and low order schemes.

Contribution

It introduces a novel filtering approach for DG discretizations of hyperbolic equations, integrating mesh adaptation to improve solution quality.

Findings

Effective reduction of oscillations near discontinuities.

Maintains high order accuracy in smooth regions.

Implemented within the deal.II library framework.

Abstract

We introduce a filtering technique for Discontinuous Galerkin approximations of hyperbolic problems. Following an approach already proposed for the Hamilton-Jacobi equations by other authors, we aim at reducing the spurious oscillations that arise in presence of discontinuities when high order spatial discretizations are employed. This goal is achieved using a filter function that keeps the high order scheme when the solution is regular and switches to a monotone low order approximation if it is not. The method has been implemented in the framework of the $deal.II$ numerical library, whose mesh adaptation capabilities are also used to reduce the region in which the low order approximation is used. A number of numerical experiments demonstrate the potential of the proposed filtering technique.