Shock Regularization with Smoothness-Increasing Accuracy-Conserving Dirac-Delta Polynomial Kernels

TL;DR

This paper introduces a novel spectral filtering method using polynomial kernels that approximate delta functions, effectively regularizing discontinuities in hyperbolic conservation laws while maintaining high-order accuracy.

Contribution

It presents a new smoothness-increasing accuracy-conserving filter based on Dirac-delta polynomial kernels for spectral collocation methods.

Findings

Regularizes discontinuities effectively

Preserves high-order resolution in test cases

Applicable to 1D and 2D hyperbolic equations

Abstract

A smoothness-increasing accuracy conserving filtering approach to the regularization of discontinuities is presented for single domain spectral collocation approximations of hyperbolic conservation laws. The filter is based on convolution of a polynomial kernel that approximates a delta-sequence. The kernel combines a $k^{th}$ order smoothness with an arbitrary number of ${m}$ zero moments. The zero moments ensure a $m^{th}$ order accurate approximation of the delta-sequence to the delta function. Through exact quadrature the projection error of the polynomial kernel on the spectral basis is ensured to be less than the moment error. A number of test cases on the advection equation, Burger's equation and Euler equations in 1D and 2D shown that the filter regularizes discontinuities while preserving high-order resolution