Algebraically constrained finite element methods for hyperbolic problems with applications in geophysics and gas dynamics
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Date
2022
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Abstract
The research conducted in this thesis is focused on property-preserving discretizations of hyperbolic partial differential equations.
Computational methods for solving such problems need to be carefully designed to produce physically meaningful numerical solutions.
In particular, approximations to some quantities of interest should satisfy local and global discrete maximum principles.
Moreover, numerical methods need to obey certain conservation relations, and convergence of approximations to the physically relevant exact solution should be ensured if multiple solutions may exist.
Many algorithms based on the aforementioned design principles fall into the category of algebraic flux correction (AFC) schemes.
Modern AFC discretizations of nonlinear hyperbolic systems express approximate solutions as convex combinations of intermediate states and constrain these states to be admissible.
The main focus of our work is on monolithic convex limiting (MCL) strategies that modify spatial semi-discretizations in this way.
Contrary to limiting approaches of predictor-corrector type, their monolithic counterparts are well suited for transient and steady problems alike.
Further benefits of the MCL framework presented in this thesis include the possibility of enforcing entropy stability conditions in addition to discrete maximum principles.
Using the AFC methodology, we transform finite element discretizations into property-preserving low order methods and perform flux correction to recover higher orders of accuracy without losing any desirable properties.
The presented methods produce physics-compatible approximations, which exhibit excellent shock capturing capabilities.
One novelty of this work is the tailor-made extension of monolithic convex limiting to the shallow water equations with a nonconservative topography term.
Our generalized MCL schemes are entropy stable, positivity preserving, and well balanced in the sense that lake at rest equilibria are preserved.
Another desirable property of numerical methods for the shallow water equations is the capability to handle wet-dry transitions properly.
We present two new approaches to dealing with this issue.
To corroborate our computational results with theoretical investigations, we perform numerical analysis for property-preserving discretizations of the time-dependent linear advection equation.
In this context, we prove stability and derive an a~priori error estimate in the semi-discrete setting.
We also compare the monolithic convex limiting strategy to two representatives of related flux-corrected transport algorithms.
Another highlight of this thesis is the chapter on MCL schemes for arbitrary order discontinuous Galerkin (DG) discretizations.
Building on algorithms developed for continuous Lagrange and Bernstein finite elements, we extend our MCL schemes to the high order DG setting.
This research effort involves the design of new AFC tools for numerical fluxes that appear in the DG weak formulation.
Our limiting strategy for DG methods exploits the properties of high order Bernstein polynomials to construct sparse discrete operators leading to compact-stencil nonlinear approximations.
The proposed numerical methods are applied to various hyperbolic problems.
Scalar equations are considered mainly for testing purposes and to simplify numerical analysis.
Besides the shallow water system, we study the Euler equations of gas dynamics.
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Keywords
Property-preserving methods, Algebraic flux correction, Limiters, Finite elements, Hyperbolic PDEs