Finite Difference Methods for Wave Dominated Problems

Sammanfattning: Wave models are an important class of models that describe many diverse phenomena such as sound waves, fluid flow, and quantum mechanics. These models are often described mathematically as partial differential equations (PDE). Often these equations do not admit solutions on closed-form and then the only option to study them is numerical methods. These numerical methods must be robust, accurate, and efficient. For spatial discretizations, it is known that higher-order finite-difference methods are efficient, but they often complicate achieving robustness.In this thesis, we focus on high-order finite-difference methods for solving these wave-dominated PDEs. We use the summation-by-parts (SBP) framework together with simultaneous approximation terms (SAT) for the boundary conditions to prove stability and robustness. This results in efficient numerical methods that are known to converge to the correct solution.The work in this thesis aims to broaden the scope of these finite-difference methods in different ways. In Paper I and III the framework is extended to two dispersive wave equations, solving challenges arising in both the spatial discretization as well as the time integration. The geometric flexibility of the methods is enhanced for the linear Euler equation and the Schrödinger equation in Paper II and VI by studying both stationary and time-dependent curvilinear grids. Paper IV shows how to use the framework to combine two models to describe and simulate a coupled physical system. In paper V we address a deficiency in the methodology around non-matching grids, showing a way to improve the accuracy and get faster convergence.