Adaptive convolution quadrature for the wave equation

PI: María López Fernández

Abstract: The efficient approximation of time domain boundary integral equations has become a very active field of research in the last years. The applications in engineering are many since the scattering of both acoustic and electromagnetic waves in unbounded exterior domains can be modeled in this way. From the computational point of view the main bottleneck of these models comes from space-dependent delays in time which appear in the resulting retarded integral equations. The main concerns in this setting are the computational cost and the memory requirements. A popular direct discretization method for these retarded integral equations in the time domain is given by discontinuous Galerkin methods. However, a drawback of this method is that, due to the coupling of space and time, integrals over the discrete light cone with the boundary element mesh have to be evaluated, which is complicated. An alternative approach is Lubich´s Convolution Quadrature, which approximates the retarded integral equations in the Laplace domain. The current state of the art for the convolution quadrature does not allow conceptually for variable time stepping. In fact the inclusion of adaptivity in time to approximate time-domain boundary integral equations is a difficult issue. The major goal of this project is the development of adaptive, fast and potentially memory-saving methods for the boundary integral formulation of the wave equation, which we choose as our model problem.