Discontinuous Galerkin method for linear wave equations involving derivatives of the Dirac delta distribution

Abstract

Linear wave equations sourced by a Dirac delta distribution $\delta (x)$ and its derivative(s) can serve as a model for many different phenomena. We describe a discontinuous Galerkin (DG) method to numerically solve such equations with source terms proportional to ${\partial }^n\delta /\partial x^n$. Despite the presence of singular source terms, which imply discontinuous or potentially singular solutions, our DG method achieves global spectral accuracy even at the source's location. Our DG method is developed for the wave equation written in fully first-order form. The first-order reduction is carried out using a distributional auxiliary variable that removes some of the source term's singular behavior. While this is helpful numerically, it gives rise to a distributional constraint. We show that a time-independent spurious solution can develop if the initial constraint violation is proportional to $\delta (x)$. Numerical experiments verify this behavior and our scheme's convergence properties by comparing against exact solutions.

arXiv comment15 pages; 4 figures. Version 2 fixes minor typos

References in corpus (10)

• The motion of point particles in curved spacetime
• Towards adiabatic waveforms for inspiral into Kerr black holes: I. A new model of the source for the time domain perturbation equation
• A Finite Element Computation of the Gravitational Radiation emitted by a Point-like object orbiting a Non-rotating Black Hole
• Discontinuous Galerkin method for computing gravitational waveforms from extreme mass ratio binaries
• Pseudospectral Collocation Methods for the Computation of the Self-Force on a Charged Particle: Generic Orbits around a Schwarzschild Black Hole
• A time-domain fourth-order-convergent numerical algorithm to integrate black hole perturbations in the extreme-mass-ratio limit
• A GPU-accelerated mixed-precision WENO method for extremal black hole and gravitational wave physics computations
• Persistent junk solutions in time-domain modeling of extreme mass ratio binaries
• A spectral collocation approximation for the radial-infall of a compact object into a Schwarzschild black hole
• Simulations of Extreme-Mass-Ratio Inspirals Using Pseudospectral Methods

Cited by in corpus (1)

• Superconvergent Discontinuous Galerkin Method for the Scalar Teukolsky Equation on Hyperboloidal Domains: Efficient Waveform and Self-Force Computation