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Waves manifest in many areas of physics, ranging from large-scale seismic waves in geophysics down to particle descriptions in quantum physics. Wave propagation may often be described mathematically by partial differential equations (PDE). Unfortunately, analytical solutions to PDEs are in many cases notoriously difficult to obtain. For this reason, one turns to approximate solutions obtained through numerical methods implemented as computer algorithms. In order for a numerical method to be useful in predictive simulations, it should be stable and accurate. Stability of the method ensures that small errors in the approximation do not grow exponentially. Accuracy together with stability ensures that increased resolution in the simulation results in decreased error in the approximation. The numerical methods considered in this thesis are finite difference methods satisfying a summation-by-parts (SBP) property. Finite difference methods are well suited for wave propagation problems in that they provide high accuracy at low computational cost. The SBP property additionally facilitates the construction of provably stable high-order accurate approximations.

This thesis continues the development of SBP finite difference methods for wave propagation problems. Paper I presents a finite difference method for modeling induced seismicity, i.e., earthquakes caused by human activity. Paper II develops a high-order accurate finite difference method for shock waves modeled by scalar conservation laws. In Paper III, new SBP finite difference operators with increased accuracy and efficiency for surface and interface waves are derived. In Papers IV - V numerical methods for inverse problems governed by wave equations are considered, where unknown model parameters are reconstructed by fitting the numerical solution to known data. Specifically, Paper IV presents a method for acoustic shape optimization, while Paper V presents an inversion method for frictional parameters used in earthquake modeling.

Partial differential equations appear in mathematical models that describe a wide range of physical phenomena, such as sound pressure waves in the air, the vibrations of solid structures, and the flow of fluids. Unfortunately, most of these problems can not be solved analytically using pen and paper. Instead, we turn to numerical methods and computer simulations to obtain approximate solutions. In this thesis, the focus is on high-order accurate finite difference methods for wave propagation and fluid dynamical problems. High-order finite difference methods are conceptually simple to design and implement efficiently on modern computers. However, special care must be taken close to boundaries to obtain robust and stable schemes. In this thesis, a class of finite difference operators with summation-by-parts (SBP) properties is used. These operators satisfy a discrete equivalent to intergration-by-parts which, when the boundary conditions are correctly imposed, enables a stability proof for the discretized scheme. Two such methods for imposing boundary conditions are studied and compared in the thesis, the simultaneous-approximation-term (SAT) method and the projection (P) method.

In Paper I a high-order accurate finite difference discretization of the incompressible Navier-Stokes equations is presented, where the projection method is found to be more suitable for wall boundary conditions. In Paper II the SBP-SAT and SBP-P methods are compared for boundary and interface conditions to the dynamic beam equation and the dynamic Kirchoff-Love plate equation. A new SBP-P and hybrid SBP-P-SAT method is developed for non-conforming interface conditions to the second-order wave equation in Paper III. In Paper IV shape optimization problems constrained by the second-order wave equation are solved using high-order SBP-P-SAT finite difference discretizations. Theoretical aspects of the projection method are discussed in Paper V. In Paper VI SBP operators defined on Gauss-Lobatto quadrature points are used to derive an efficient and robust scheme for the Laplacian on complex geometries.