Advanced BEM for wave propagation in the time domain

Waves are one of the fundamental phenomena in the natural world. From acoustics, to optics, to quantum mechanics, wave-like effects occur. This makes the numerical simulations of the wave equation and its cousins like the Schrödinger equation important for technological and physical applications. Instead of simulating waves on the whole ocean, one is often interested in only a much smaller subdomain. This raises the question on what to do at the new artificial boundary. A standard approach would give unnatural reflections which would spoil the accuracy of the numerical simulation. The basis of the project is the so called boundary element method, in which instead of solving an equation on a possibly infinite volume, the equation is reduced to the boundary of the object of interest. This not only reduces the domain to the finite domain of interest but also reduces the dimension of the problem, from the three-dimensional volume to a two-dimensional surface. For treating the time-dependent aspect of the problem, we use the a method known as convolution quadrature. The goal of this project is, on the one hand to better understand the behavior of existing algorithms for such wave propagation problems, and then to design new and improved algorithms which give a better accuracy for the simulations while decreasing the computational effort needed. On the other hand, we want to apply known algorithms to a wider class of problems, most notably to the world of nonlinear wave equations. In addition to the theoretical work, the project also has a software-engineering component, wherein a new open-source library is developed. The goal of this library is to make both, the state of the art algorithms already proposed in the literature, as well as the new approaches from this project, available to the wider research community.