High Performance Algorithms for Time Dependent Acoustic Obstacle Scattering

This paper presents a highly parallelizable numerical method to solve time dependent acoustic obstacle scattering problems. The method proposed is a generalization of the "operator expansion method" developed by M.C. Recchioni, F. Zirilli (2003, SIAM J. Sci. Comput. 25 1158-1186). Using a Fourier transform with respect to time the time dependent scattering problem for the wave equation considered is transformed in an exterior boundary value problem for the Helmholtz equation depending on a parameter. The "operator expansion method" is used to solve the exterior problems for the Helmholtz equation and reduces, via a perturbative approach, the solution of each exterior problem to the solution of a sequence of systems of first kind integral equations defined on a suitable "reference surface". The numerical solution of these systems of integral equations is challenging when scattering problems involving realistic obstacles and wavelengths small compared to the characteristic dimensions of the obstacles are solved. Let R_T be the ratio between the characteristic dimension of the obstacle and the wavelength considered. In the numerical experiments presented the greater ratios R_T considered range in the interval [13, 60]. This implies the use of high dimensional vector spaces to approximate satisfactorily the corresponding systems of integral equations. That is each system of integral equations in the sequence mentioned above after being discretized to be solved numerically becomes a large system of linear equations. Represented on a generic base the matrices that approximate the integral operators after the discretization will be dense matrices. In this paper a new way of using the wavelet transform and new bases of wavelets are introduced and a version of the operator expansion method is developed that constructs directly element by element the coefficient matrix of the sparse linear systems that approximate in the wavelet basis considered the system- s of integral equations. The resulting numerical method, using affordable computing resources, is able to deal with realistic acoustic scattering problems solving large (sparse) linear systems (up to approximately 5-10⁵ (real) unknowns and equations in the numerical experience shown here). The computation of the elements of the coefficient matrices of the linear systems considered and several other aspects of the numerical algorithm proposed are highly parallelizable allowing the development of an efficient solver for exterior boundary value problems for the Helmholtz equation. Several numerical experiments involving realistic obstacles and "small" wavelengths are proposed. When possible the quantitative character of the numerical results obtained is established. To evaluate the performance of the proposed algorithm on parallel computing facilities appropriate speed up factors are introduced and evaluated. Some animations and virtual reality applications relative to these numerical experiments can be seen in the website: http://www. econ. univpm. it/recchioni/wl2l. A general reference for the work done on scattering by the author and its coworkers is http://www. econ. univpm. it/recchioni/\´scattering.