2025 Theses Doctoral

Discontinuous Galerkin Methods for Nonlinear and Nonlocal Wave Equations

Zhou, Yin

This dissertation develops and analyzes discontinuous Galerkin (DG) methods for two wave propagation problems: the nonlinear Schrodinger equation with wave operator (NLSW) and the nonlocal wave equation (NLW). These models arise in diverse physical and engineering contexts ranging from plasma physics to peridynamic solid mechanics, and pose significant numerical challenges due to their dispersive, nonlinear, and nonlocal behavior. Efficient numerical algorithms for these problems are essential for accurately simulating phenomena such as soliton dynamics and long-range interactions.

For the NLSW model, we propose an energy-based DG formulation that preserves discrete energy properties and admits optimal convergence in the L2 norm. A key component of the method is the introduction of an auxiliary variable along with the careful design of mesh-independent numerical fluxes. We establish rigorous stability and error estimates for the semi-discrete scheme. To handle the time dimension, we employ a strong-stability-preserving Runge–Kutta (SSPRK) scheme, ensuring robust and accurate temporal integration. The method is validated through extensive numerical simulations in one and two spatial dimensions.

For the NLW model, we construct and analyze a DG method that accommodates spatial nonlocality and is capable of capturing the asymptotic transition to local models. We discretize in time via a Crank–Nicolson scheme, combining second-order accuracy with favorable stability properties. The method preserves energy at the fully-discrete level and exhibits optimal convergence rates for a broad class of nonlocal kernels. We further establish asymptotic compatibility, ensuring that the scheme recovers the classical wave equation in the vanishing-horizon limit.

In addition to standard numerical analysis of the schemes, we demonstrate, through extensive numerical simulations, the accuracy and effectiveness of the proposed methods. This dissertation offers some new insights on developing advanced DG-type methods for nonlinear and nonlocal wave equations, especially in second order form.