# High Order Edge Sensors with $\ell^1$ Regularization for Enhanced Discontinuous Galerkin Methods

## TL;DR

This paper introduces a novel high order edge sensor with $$ regularization for discontinuous Galerkin methods, improving the handling of troubled regions in hyperbolic conservation laws efficiently.

## Contribution

It develops a high order edge sensor using polynomial annihilation and applies $$ regularization selectively to troubled cells in DG methods, enhancing accuracy and efficiency.

## Key findings

- Effective detection of troubled elements with the edge sensor.
- Improved solution accuracy in discontinuous regions.
- Efficient implementation using ADMM for $$ optimization.

## Abstract

This paper investigates the use of $\ell^1$ regularization for solving hyperbolic conservation laws based on high order discontinuous Galerkin (DG) approximations. We first use the polynomial annihilation method to construct a high order edge sensor which enables us to flag troubled elements. The DG approximation is enhanced in these troubled regions by activating $\ell^1$ regularization to promote sparsity in the corresponding jump function of the numerical solution. The resulting $\ell^1$ optimization problem is efficiently implemented using the alternating direction method of multipliers. By enacting $\ell^1$ regularization only in troubled cells, our method remains accurate and efficient, as no additional regularization or expensive iterative procedures are needed in smooth regions. We present results for the inviscid Burgers' equation as well as a nonlinear system of conservation laws using a nodal collocation-type DG method as a solver.