Regularization of singularities in the weighted summation of Dirac-delta functions for the spectral solution of hyperbolic conservation laws?

TL;DR

This paper introduces a regularization technique for singular source terms in hyperbolic conservation laws using convolution with high-order Dirac-delta approximations, improving spectral solution accuracy.

Contribution

It develops an optimal scaling parameter for Dirac-delta regularization, enhancing spectral methods' accuracy near singularities in hyperbolic PDEs.

Findings

Regularization improves spectral solution accuracy near singularities.

Optimal scaling parameter achieves high-order accuracy in smooth regions.

Numerical tests confirm enhanced accuracy with the proposed method.

Abstract

Singular source terms expressed as weighted summations of Dirac-delta functions are regularized through approximation theory with convolution operators. We consider the numerical solution of scalar and one-dimensional hyperbolic conservation laws with the singular source by spectral Chebyshev collocation methods. The regularization is obtained by convolution with a high-order compactly supported Dirac-delta approximation whose overall accuracy is controlled by the number of vanishing moments, degree of smoothness and length of the support (scaling parameter). An optimal scaling parameter that leads to a high-order accurate representation of the singular source at smooth parts and full convergence order away from the singularities in the spectral solution is derived. The accuracy of the regularization and the spectral solution is assessed by solving an advection and Burgers equation with smooth initial data. Nu- merical results illustrate the enhanced accuracy of the spectral method through the proposed regularization.