Computation of Acoustic Wave Propagation Under Water

Sammanfattning: In this thesis we look at acoustic wave propagation under water. We look in particular at waves generated by a point source and what happens with the propagation when we model the bottom as flat or as curvilinear. We assume the source to be working at a certain frequency and therefore we model this problem by solving the Helmholtz equation. Since Helmholtz equation has some unwanted numerical properties we are interested in finding new numerical methods that could accelerate the solver. In this thesis we use the Waveholtz iteration, which solves Helmholtz equation by connecting it to the time-dependent wave equation. We use finite differences and the SBP-SAT method to approximate the spatial problem numerically and for modelling the sea bottom we use curvilinear coordinates.  To compare the Waveholtz iteration we also solve Helmholtz equation with a naive solver. The naive solver consists of approximating the equation with finite differences and then solving the linear system of equation by some iterative solver, which for our tests will be GMRES. The results show that the Waveholtz iteration converges in less iterations than our naive solver. It also shows that the number of iterations stays unchanged when changing our discretization, which otherwise is a big problem for our naive solver. This allows us to increase the accuracy of our numerical solution without changing the computation time too much.  We show that the number of iterations increases according to theory for an increasing frequency, and that for open problems we even see a smaller increase. For certain resonant frequencies in Helmholtz equation we do not expect the Waveholtz iteration to converge. In the neighbourhood of these frequencies the convergence becomes slow and we need many iterations for a solution of a certain accuracy. By reformulating the Waveholtz iteration as a Krylov solution we can see that resonances in Helmholtz equation have a smaller impact of the convergence.