A convolution quadrature Galerkin boundary element method for the exterior Neumann problem of the wave equation

Abstract

SUMMARYThe numerical solution of the Neumann problem of the wave equation on unbounded three-dimensional domains is calculated using the convolution quadrature method for the time discretization and a Galerkin boundary element method for the spatial discretization. The mathematical analysis that has been built up for the Dirichlet problem is extended and developed for the Neumann problem, which is important for many modelling applications. Numerical examples are then presented for one of these applications, modelling transient acoustic radiation.

Fast convolution quadrature for the wave equation in three dimensions

“…With the choice of the parameter L as in this section also a part of the far-field is reused, and hence fewer multipole-to-local translations have to be done when computing the FMM accelerated matrix-vector product. Due to the use of the parameter d as in (36), the smallest distance between the admissible leaves of the 'far-field' block-cluster tree (i.e. L + D n ,F in (34)) is larger than the smallest distance between admissible leaves of the full block-cluster tree (since small close clusters are contained in the 'near-field' block-cluster tree).…”

“…Compared to multistep convolution quadrature, Runge-Kutta convolution quadrature has low dissipation and dispersion, see [34,35] for a quantifiable definition and analysis of these properties, as well as numerical experiments in [29]. Recent works [36,37] provide the analysis of multistep convolution quadrature combined with the Galerkin discretization for the scattering by a sound-hard obstacle, as well as suggest the procedure of the reduced convolution weight computation. In [38] the convolution quadrature formulation for the Maxwell equations was studied analytically and numerically.…”

“…In parentheses we show the CPU time needed to solve the system of equations after all the matrices were constructed. Here h is the time step, N is the number of time steps, M is size of the spatial discretization and L is as in(36).…”

Overresolving in the Laplace Domain for Convolution Quadrature Methods

“…In recent years they have seen tremendous interest for the solution of exterior time-domain scattering problems via boundary integral equation formulations, see e.g. [7,11,2,8,12,21]. The application to Maxwell problems is discussed in [13,1].…”

Stable numerical coupling of exterior and interior problems for the wave equation

“…The standard discretization in space is by boundary elements (in their Galerkin or collocation variants). Two classes of discretizations in time are known to yield guaranteed stability: the space-time Galerkin approach (Ha Duong [13], Ha Duong, Ludwig & Terrasse [14]) and convolution quadrature (Lubich [21] and more recently Hackbusch, Kress & Sauter [15], Banjai & Sauter [7], Banjai [5], Banjai, Lubich & Melenk [6], Chappell [9], Chen, Monk, Wang & Weile [24], Monegato, Scuderi & Stanić [23]). Here we use convolution quadrature for time discretization of the boundary integrals.…”