High-Order Discontinuous Galerkin Methods for One Dimensional Flows and Optimal Control

Leonor Figueiredo Sopas Barardo Nobre

Key information

Authors:

Leonor Figueiredo Sopas Barardo Nobre (Leonor Figueiredo Sopas Barardo Nobre)

Supervisors:

João Carlos de Campos Henriques (João Carlos de Campos Henriques)

Published in

July 15, 2025

Abstract

This dissertation investigates the integration of the Discontinuous Galerkin Method (DGM) in Optimal Control and Hyperbolic Systems, focusing on accuracy, stability, and shock capturing. The first part addresses bang-bang optimal control problems governed by ordinary differential equations with discontinuous right-hand sides. It analyses the degradation of high-order numerical accuracy caused by switching time position and finite precision arithmetic. A time-dependent optimal control problem and a convection-dominated shallow water PDE are studied to evaluate the effect of discontinuities. Results show that accuracy significantly decreases when discontinuities lie inside elements, regardless of arithmetic precision or polynomial order. Furthermore, finite digit precision, such as double precision, limits the achievable mesh convergence order as polynomial degree increases. The second part presents a DG solver for hyperbolic PDEs, implemented in Python, targeting engineering problems governed by Shallow Water and Euler equations. The solver features efficient matricial implementation with modal and nodal formulations, entropy-consistent fluxes, and shock stabilization using modal decay sensors and localized artificial viscosity. A mathematical analysis of energy stability, convergence, and superconvergence for linear convection-diffusion systems is provided. Numerical experiments validate the approach: shallow water simulations with discontinuous bathymetry achieve optimal convergence only when mesh interfaces align with discontinuities; transcritical flows with hydraulic jumps are stabilized via adaptive viscosity, resolving shocks without spurious oscillations; compressible nozzle flows demonstrate entropy-stable DG performance in smooth and discontinuous regimes. This work bridges theory and computation, offering valuable insights for fluid dynamics, control engineering, and high-order numerical methods.