A High-Order Dirac-Delta Regularization with Optimal Scaling in the Spectral Solution of One-Dimensional Singular Hyperbolic Conservation Laws

A regularization technique based on a class of high-order, compactly supported piecewise polynomials is developed that regularizes the time-dependent, singular Dirac-delta sources in spectral approximations of hyperbolic conservation laws. The regularization technique provides higher-order accuracy away from the singularity. A theoretical criterion that establishes a lower bound on the support length (optimal scaling) has to be satisfied to achieve optimal order of convergence. The optimal scaling parameter has been shown to be, instead of a fixed constant value, a function of the smoothness of the compactly supported piecewise polynomial and the number of vanishing moment of the polynomial when integrated with respect to mononomial of degree up to some order (moment). The effectiveness of the criterion is illustrated in the solutions of a linear and a nonlinear (Burgers) scalar hyperbolic conservation law with a singular source, as well as the nonlinear Euler equations with singular sources, a system of hyperbolic conservation laws governing compressible fluid dynamics with shocks and particles. A Chebyshev collocation method (spectral) discretizes the spatial derivatives in the scalar equation tests. A multidomain hybrid spectral-WENO method discretizes the Euler equations. The WENO scheme captures shocks and high gradients in the gas flow, whereas the spectral discretization applies in all other subdomains including the regularization of the singular source. In the advection and Burgers problems, the convergence order of the numerical solution follows the asymptotic behavior of the singular source approximation. The multidomain hybrid scheme is in excellent agreement with the computations that are based on a WENO discretization only.