Monolithic multigrid methods for high-order discretizations of time-dependent PDEs

Abu-Labdeh, Razan (2023) Monolithic multigrid methods for high-order discretizations of time-dependent PDEs. Doctoral (PhD) thesis, Memorial University of Newfoundland.

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A currently growing interest is seen in developing solvers that couple high-fidelity and higher-order spatial discretization schemes with higher-order time stepping methods for various time-dependent fluid plasma models. These problems are famously known to be stiff, thus only implicit time-stepping schemes with certain stability properties can be used. Of the most powerful choices are the implicit Runge-Kutta methods (IRK). However, they are multi-stage, often producing a very large and nonsymmetric system of equations that needs to be solved at each time step. There have been recent efforts on developing efficient and robust solvers for these systems. We have accomplished this by using a Newton-Krylov-multigrid approach that applies a multigrid preconditioner monolithically, preserving the system couplings, and uses Newton’s method for linearization wherever necessary. We show robustness of our solver on the single-fluid magnetohydrodynamic (MHD) model, along with the (Navier-)Stokes and Maxwell’s equations. For all these, we couple IRK with higher-order (mixed) finiteelement (FEM) spatial discretizations. In the Navier-Stokes problem, we further explore achieving more higher-order approximations by using nonconforming mixed FEM spaces with added penalty terms for stability. While in the Maxwell problem, we focus on the rarely used E-B form, where both electric and magnetic fields are differentiated in time, and overcome the difficulty of using FEM on curved domains by using an elasticity solve on each level in the non-nested hierarchy of meshes in the multigrid method.

Item Type: Thesis (Doctoral (PhD))
Item ID: 16162
Additional Information: Includes bibliographical references
Keywords: multigrid methods, Runge-Kutta, numerical methods, Vanka relaxation, incompressible flow
Department(s): Science, Faculty of > Mathematics and Statistics
Date: September 2023
Date Type: Submission
Digital Object Identifier (DOI):
Library of Congress Subject Heading: Multigrid methods (Numerical analysis); Runge-Kutta formulas; Fluid dynamics--Mathematical models; Plasma dynamics--Mathematical models; Differential equations, Partial; computer science--Mathematics

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