# Multi-element SIAC filter for shock capturing applied to high-order discontinuous Galerkin spectral element methods

## TL;DR

This paper introduces a multi-element SIAC filter for high-order discontinuous Galerkin spectral element methods, enhancing shock capturing while maintaining accuracy in simulations of conservation laws like Euler and MHD equations.

## Contribution

It develops a novel multi-element SIAC filtering technique that adaptively captures shocks in high-order DG methods without sacrificing accuracy.

## Key findings

- Effective shock capturing in 2D Euler and MHD tests
- Maintains high-order accuracy near discontinuities
- Applicable to general systems of conservation laws

## Abstract

We build a multi-element variant of the smoothness increasing accuracy conserving (SIAC) shock capturing technique proposed for single element spectral methods by Wissink et al. (B.W. Wissink, G.B. Jacobs, J.K. Ryan, W.S. Don, and E.T.A. van der Weide. Shock regularization with smoothness-increasing accuracy-conserving Dirac-delta polynomial kernels. Journal of Scientific Computing, 77:579--596, 2018). In particular, the baseline scheme of our method is the nodal discontinuous Galerkin spectral element method (DGSEM) for approximating the solution of systems of conservation laws. It is well known that high-order methods generate spurious oscillations near discontinuities which can develop in the solution for nonlinear problems, even when the initial data is smooth. We propose a novel multi-element SIAC filtering technique applied to the DGSEM as a shock capturing method. We design the SIAC filtering such that the numerical scheme remains high-order accurate and that the shock capturing is applied adaptively throughout the domain. The shock capturing method is derived for general systems of conservation laws. We apply the novel SIAC filter to the two-dimensional Euler and ideal magnetohydrodynamics (MHD) equations to several standard test problems with a variety of boundary conditions.