Gryphon - a Module for Time Integration of Partial Differential Equations in FEniCS

Abstract

This thesis aims to implement time integrators in the FEniCS framework. More specifically, the thesis focuses on selecting suitable time integrators, implement these and verify that the implementation works by applying them to various relevant test problems. This work resulted in a module for FEniCS, named Gryphon. The thesis is divided into four parts.The first part builds a theoretical framework which will motivate why singly diagonally implicit Runge-Kutta methods with an explicit first stage (ESDIRKs) should be considered for solving stiff ordinary differential equations (ODEs). It will also be shown how an ESDIRK method can be utilized to solve time dependent partial differential equations (PDEs) by solving the semidiscretized system arising from first applying a finite element method. We will restrict our attention to PDEs which either give rise to a pure ODE system or a DAE (differential-algebraic equation) system of index 1.The second part discusses the implementation of Gryphon, focusing on why such a module is useful and how the source code is structured.The third part is devoted to numerical experiments on the ESDIRK solvers implemented in Gryphon. The experiments will establish convergence and give some run-time statistics for various ESDIRK schemes. We will also see that L-stability is a favorable trait when working with stiff equations, by comparing an ESDIRK method to the trapezoidal rule. It will also be verified that the step size selectors implemented in Gryphon behaves as expected. As test problems we consider the heat equation, the Fisher-Kolmogorov equation, the Gray-Scott equations, the Fitzhugh-Nagumo equations and the Cahn-Hilliard equations.The fourth part is a user manual for Gryphon. All the parameters which can be changed by the user are explained. The manual also includes example code for solving the heat equation, the Gray-Scott equations and the Cahn-Hilliard equation, to get the reader starting on solving their own problems.