Finite Difference Methods for Time-Dependent Wave Propagation Problems

Sammanfattning: Wave propagation models are essential in many fields of physics, such as acoustics, electromagnetics, and quantum mechanics. Mathematically, waves can be described by partial differential equations (PDEs).  In most cases, exact solutions to wave-dominated PDEs are nearly impossible to derive. Hence, numerical methods and computer simulations are necessary when studying wave propagation problems. It is well known that high-order finite difference methods have good properties for solving wave propagation problems efficiently. However, for time-dependent wave dominated problems that include boundaries, it has historically been challenging to construct stable discretizations with these types of methods.In this thesis, high-order summation-by-parts (SBP) finite difference methods for time-dependent wave propagation problems from acoustics and quantum mechanics are studied. The SBP property combined with a simultaneous approximation term (SAT) allows the construction of provably stable finite difference schemes. However, real-world problems often include complicated features such as complex geometries, non-linearities, or non-smooth data that may ruin the accuracy of SBP-SAT. In this thesis, high-order SBP-SAT schemes are derived for problems including two such features: moving boundaries and point sources.In Paper I, two stable SBP-SAT schemes for the linearized Euler equations are derived. These equations govern acoustic wave propagation in the atmosphere. The schemes allow point sources, complex geometries, and varying material coefficients. Numerical schemes for problems on deforming domains are developed for the time-dependent Schrödinger equation in Paper II and the 1D Euler equations in Paper III. In Paper IV, a robust numerical discretization for point sources that move in the domain is presented. Finally, in paper V implicit SBP operators are derived. These operators have good properties for long-time simulations of wave propagation problems that include high frequencies.