Summation-by-Parts Finite Difference Methods for Wave Propagation and Earthquake Modeling

Abstract: 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.