Date of Award

2025

Document Type

Open Access Dissertation

Degree Name

Doctor of Philosophy in Mathematical Sciences (PhD)

Administrative Home Department

Department of Mathematical Sciences

Advisor 1

Yang Yang

Committee Member 1

Zhengfu Xu

Committee Member 2

Alexander E. Labovsky

Committee Member 3

Chunpei Cai

Abstract

This dissertation is composed of four chapters in which we will closely examine the high order bound preserving discontinuous Galerkin methods for solving partial differential equations, specifically non-equilibrium chemical reacting flows and Euler equations under gravitational fields. A shared requirement between the two is the necessity for positive values of both density and pressure. Due to this physical nature of the two systems, constructing a positivity preserving scheme become very essential in our research.

For non-equilibrium flows where multi-reactions and multi-species are involved, we are also required to keep the bounds of the mass fraction of each species in between 0 and 1. In this case, the positivity preserving technique should be applied to each species to preserve the lower bound 0. Then enforce the summation of the mass fraction be 1 by using consistent flux and conservative time integration for maintaining the upper bound 1. We apply the Patankar time integration. This is a conservative implicit method targeted stiff source term to reduce the computational cost. We develop the bound-preserving discontinuous Galerkin method coupled with the Patankar time integration for the purpose of conservation, bound preserving, and efficiency. The bound preserving DG and Patankar time integration require $Q^k$ polynomials on rectangular meshes in two dimensional space to match the degree of freedom. The reason is that Patankar can keep the positivity of the pre-selected point-values of the target variables but the positivity-preserving technique for DG method requires positive numerical approximations at the cell interfaces. Additionally, slope limiters are applied to ensure the positivty of the numerical solutions.

The Euler equations under gravitational fields admit steady state solutions. Therefore, besides being positivity preserving, we should also emphasize on the ability of the numerical scheme to yield steady-state solutions when at equilibrium and to catch the small perturbations when they appear. The well-balanced scheme should be used for this purpose or else the truncation error will be built up and destroy the robustness. We develop high order positivity-preserving well-balanced discontinuous Galerkin methods with Lax-Friedrich fluxes. The difficulty of such scheme is the compatibility of the PP DG and WB. One needs the penalty term in the flux to be large and the other needs it to be zero. The solution to this concern is to use the relationship between numerical fluxes to obtain a new penalty term which not only guarantees the positivity but also turns zero at the steady state. For the two studies, numerical experiments are given to demonstrate the performance of both schemes.

Download

Share

COinS