High-Order Nystr\"om/Convolution-Quadrature Solution of Time-Domain Scattering from Closed and Open Lipschitz Boundaries with Dirichlet and Neumann Boundary Conditions

TL;DR

This paper develops high-order convolution quadrature methods combined with Nyström discretizations to accurately solve 2D wave scattering problems with complex boundary conditions, demonstrating up to fifth-order temporal convergence.

Contribution

It introduces a novel combination of Nyström-based boundary integral discretizations with convolution quadrature methods for high-order time integration in wave scattering problems.

Findings

Achieves high-order (up to fifth order) convergence in time.

Validates methods on various 2D scatterers and boundary conditions.

Demonstrates effectiveness of Nyström CQ discretizations for wave equations.

Abstract

We investigate high-order Convolution Quadratures methods for the solution of the wave equation in unbounded domains in two dimensions that rely on Nystr\"om discretizations for the solution of the ensemble of associated Laplace domain modified Helmholtz problems. We consider two classes of CQ discretizations, one based on linear multistep methods and the other based on Runge-Kutta methods, in conjunction with Nystr\"om discretizations based on Alpert and QBX quadratures of Boundary Integral Equation (BIE) formulations of the Laplace domain Helmholtz problems with complex wavenumbers. We present a variety of accuracy tests that showcase the high-order in time convergence (up to and including fifth order) that the Nystr\"om CQ discretizations are capable of delivering for a variety of two dimensional scatterers and types of boundary conditions.