Lehel Banjai

Rapid Solution of the Wave Equation in Unbounded Domains

In this paper we propose and analyze a new, fast method for the numerical solution of time domain boundary integral formulations of the wave equation. We employ Lubichs convolution quadrature method for the time discretization and a Galerkin boundary element method for the spatial discretization. The coefficient matrix of the arising system of linear equations is a triangular block Toeplitz matrix. Possible choices for solving the linear system arising from the above discretization include the use of fast Fourier transform (FFT) techniques and the use of data-sparse approximations. By using FFT techniques, the computational complexity can be reduced substantially while the storage cost remains unchanged and is, typically, high. Using data-sparse approximations, the gain is reversed; i.e., the computational cost is (approximately) unchanged while the storage cost is substantially reduced. The method proposed in this paper combines the advantages of these two approaches. First, the discrete convolution (related to the block Toeplitz system) is transformed into the (discrete) Fourier image, thereby arriving at a decoupled system of discretized Helmholtz equations with complex wave numbers. A fast data-sparse (e.g., fast multipole or panel-clustering) method can then be applied to the transformed system. Additionally, significant savings can be achieved if the boundary data are smooth and time-limited. In this case the right-hand sides of many of the Helmholtz problems are almost zero, and hence can be disregarded. Finally, the proposed method is inherently parallel. We analyze the stability and convergence of these methods, thereby deriving the choice of parameters that preserves the convergence rates of the unperturbed convolution quadrature. We also present numerical results which illustrate the predicted convergence behavior.

Multistep and Multistage Convolution Quadrature for the Wave Equation: Algorithms and Experiments

We describe how a time-discretized wave equation in a homogeneous medium can be solved by boundary integral methods. The time discretization can be a multistep, Runge-Kutta, or a more general multistep-multistage method. The resulting convolutional system of boundary integral equations belongs to the family of convolution quadratures of Lubich. The aim of this work is twofold. It describes an efficient, robust, and easily parallelizable method for solving the semidiscretized system. The resulting algorithm has the main advantages of time-stepping methods and of Fourier synthesis: at each time-step, a system of linear equations with the same system matrix needs to be solved, yet computations can easily be done in parallel; the computational cost is almost linear in the number of time-steps; and only the Laplace transform of the time-domain fundamental solution is needed. The new aspect of the algorithm is that all this is possible without ever explicitly constructing the weights of the convolution quadrature. This approach also readily allows the use of modern data-sparse techniques to efficiently perform computation in space. We investigate theoretically and numerically to which extent hierarchical matrix (

Wave Propagation Problems Treated with Convolution Quadrature and BEM

\mathcal H

A Refined Galerkin Error and Stability Analysis for Highly Indefinite Variational Problems

-matrix) techniques can be used to speed up the space computation. The second aim of this article is to perform a series of large-scale 3D experiments with a range of multistep and multistage time discretization methods: the backward difference formula of order 2 (BDF2), the Trapezoid rule, and the 3-stage Radau IIA methods are investigated in detail. One of the conclusions of the experiments is that the Radau IIA method often performs overwhelmingly better than the linear multistep methods, especially for problems with many reflections, yet, in connection with hyperbolic problems, BDFs have so far been predominant in the literature on convolution quadrature.

A Multipole Method for Schwarz--Christoffel Mapping of Polygons with Thousands of Sides

Boundary element methods for steady state problems have reached a state of maturity in both analysis and efficient implementation and have become an ubiquitous tool in engineering applications. Their time-domain counterparts, however, in particular for wave propagation phenomena, still present many open questions related to the analysis of the numerical methods and their algorithmic implementation. In recent years many exciting results have been achieved in this area. In this review paper, a particular type of methods for treating time-domain boundary integral equations (TDBIE), the convolution quadrature, is described together with application areas and most recent improvements to the analysis and efficient implementation. An important attraction of these methods is their intrinsic stability, often a problem with numerical methods for TDBIE of wave propagation. Further, since convolution quadrature, though a time-domain method, uses only the kernel of the integral operator in the Laplace domain, it is widely applicable also to problems such as viscoelastodynamics, where the kernel is known only in the Laplace domain. This makes convolution quadrature for TDBIE an important numerical method for wave propagation problems, which requires further attention.

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

Recently, a refined finite element analysis for highly indefinite Helmholtz problems was introduced by the second author. We generalize the analysis to the Galerkin method applied to an abstract highly indefinite variational problem. In the refined analysis, the condition for stability and a quasi-optimal error estimate are expressed in terms of approximation properties

Numerical solution of exterior Maxwell problems by Galerkin BEM and Runge---Kutta convolution quadrature

{\cal T}(S) \approx S

Fast convolution quadrature for the wave equation in three dimensions

and

Revisiting the Crowding Phenomenon in Schwarz-Christoffel Mapping

{\cal T}(u+S) \approx S

A Trefftz polynomial space-time discontinuous Galerkin method for the second order wave equation

. Here,