Efficient Numerical Schemes for Evolution Equations with Singularities and Shocks.
Zuodong Wang: Thursday, 18th Sep 2025 – 10h30 to 12h00 Abstract: In this Thesis, we develop efficient numerical schemes for evolution partial differential equations (PDEs) with shocks and singularities. Such PDEs arise in diverse applications, including fluid dynamics, phase transition, and radiation transport. In Chapter 3, we solve the Yee–LeVeque model and its generalization using a novel numerical scheme that perturbs the stiff reaction parameter in a mesh-dependent manner. For this scheme, we establish a maximum principle and entropy inequalities without restrictive assumptions on the mesh-size or the time-step. In Chapter 4, we study the steady neutron transport equation with stiff reaction terms. By applying the edge stabilization technique, we relax some mesh constraints and develop an efficient post-processing technique to remove non-physical oscillations while preserving the positivity and conservation properties of the scheme. In Chapter 5, we focus on the Allen–Cahn equation. A fundamental phase-field model with stiff source terms. Here, we propose a nonlinear scheme for which we prove a maximum principle and derive an optimal energy-error bound with polynomial dependence on the singularity perturbation factor (or reaction parameter). In Chapter 6, we investigate the linear wave equation. Our contributions are novel $hp$-a priori and a posteriori error bounds, valid even for low-regularity solutions, and an adaptive time refinement scheme driven by our a posteriori error estimators. Numerical experiments across all chapters demonstrate the efficiency and reliability of the proposed schemes.
