Gustav Eriksson: Robust and efficient discretizations of wave-dominated problems

Abstract

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.