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

Abstract

This paper investigates the use of 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 1 regularization to promote sparsity in the corresponding jump function of the numerical solution. The resulting 1 optimization problem is efficiently implemented using the alternating direction method of multipliers. By enacting 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 for a nonlinear system of conservation laws using a nodal collocation-type DG method as a solver.

The physics of financial networks

“…Furthermore, it appears that the differentiation matrices of RBF methods encountered in time-dependent PDEs often have eigenvalues with a positive real part resulting in unstable methods; see [76]. Hence, in the presence of rounding errors, these methods are less accurate [55,75,80] and can become unstable in time unless a dissipative time integration method [64,76], artificial dissipation [22,77,39,37,73], or some other stabilizing technique [79,28,35,48,40,30,15] is used. So far, this issue was only overcome for problems which are free of BCs [64].…”

Stable high-order cubature formulas for experimental data

“…Since κ(w) = w 1 , the resulting optimization problem corresponds to 1 -minimization (which is strongly connected to compressed sensing [5,6,16,28]). Thus, the element w * is called an 1 -solution from W ⊂ R N , which is denoted as…”

“…In many case, the 1 -solution furthermore has the property of being a sparse solution; see [18,19,52] (also see [17]). In recent years, this motivated many researchers to use the 1 -norm as a surrogate for the 0 -"norm" (number of nonzero entries) [5,6,16,28]. 7 In my own implementation, I used the Matlab function minL1lin [40] to solve (14).…”

Shock capturing by Bernstein polynomials for scalar conservation laws

“…Another drawback arises from the fact that AV terms can introduce additional harsh time step restrictions, when not constructed with care, and thus decrease the efficiency of the numerical method [18,25]. Finally, we mention those methods based on order reduction [5,10], mesh adaptation [14], weighted essentially nonoscillatory (WENO) concepts [52,53], and 1 regularisation applied to high order approximations of the jump function [17]. Yet, a number of issues still remains unresolved.…”

“…In this new strategy, the approximation of u in each element may vary from usual (high-order) interpolation polynomials to a Bernstein reconstruction of the solution. Further, by employing a discontinuity sensor, here based on comparing polynomial annihilation operators [3] of increasing orders as proposed in [17], the order of the approximations is reduced to one only in elements where the solution is not smooth. For instance mesh adaptation is hence not mandatory and (shock) discontinuities can be captured without modifying the number of degrees of freedom, the mesh topology, or even the method.…”