Skip to main content

Lund University Publications

LUND UNIVERSITY LIBRARIES

Efficient Solvers for Space-Time Discontinuous Galerkin Spectral Element Methods

Versbach, Lea Miko LU (2022)
Abstract
In this thesis we study efficient solvers for space-time discontinuous Galerkin spectral element methods (DG-SEM). These discretizations result in fully implicit schemes of variable order in both spatial and temporal directions. The popularity of space-time DG methods has increased in recent years and entropy stable space-time DG-SEM have been constructed for conservation laws, making them interesting for these applications.

The size of the nonlinear system resulting from differential equations discretized with space-time DG-SEM is dependent on the order of the method, and the corresponding Jacobian is of block form with dense blocks. Thus, the problem arises to efficiently solve these huge nonlinear systems with regards to CPU... (More)
In this thesis we study efficient solvers for space-time discontinuous Galerkin spectral element methods (DG-SEM). These discretizations result in fully implicit schemes of variable order in both spatial and temporal directions. The popularity of space-time DG methods has increased in recent years and entropy stable space-time DG-SEM have been constructed for conservation laws, making them interesting for these applications.

The size of the nonlinear system resulting from differential equations discretized with space-time DG-SEM is dependent on the order of the method, and the corresponding Jacobian is of block form with dense blocks. Thus, the problem arises to efficiently solve these huge nonlinear systems with regards to CPU time as well as memory consumption. The lack of good solvers for three-dimensional DG applications has been identified as one of the major obstacles before high order methods can be adapted for industrial applications.

It has been proven that DG-SEM in time and Lobatto IIIC Runge-Kutta methods are equivalent, in that both methods lead to the same discrete solution. This allows to implement space-time DG-SEM in two ways: Either as a full space-time system or by decoupling the temporal elements and using implicit time-stepping with Lobatto IIIC methods. We compare theoretical properties and discuss practical aspects of the respective implementations.

When considering the full space-time system, multigrid can be used as solver. We analyze this solver with the local Fourier analysis, which gives more insight into the efficiency of the space-time multigrid method.

The other option is to decouple the temporal elements and use implicit Runge-Kutta time-stepping methods. We suggest to use Jacobian-free Newton-Krylov (JFNK) solvers since they are advantageous memory-wise. An efficient preconditioner for the Krylov sub-solver is needed to improve the convergence speed. However, we want to avoid constructing or storing the Jacobian, otherwise the favorable memory consumption of the JFNK approach would be obsolete. We present a preconditioner based on an auxiliary first order finite volume replacement operator. Based on the replacement operator we construct an agglomeration multigrid preconditioner with efficient smoothers using pseudo time integrators. Then only the Jacobian of the replacement operator needs to be constructed and the DG method is still Jacobian-free. Numerical experiments for hyperbolic test problems as the advection, advection-diffusion and Euler equations in several dimensions demonstrate the potential of the new approach. (Less)
Please use this url to cite or link to this publication:
author
supervisor
opponent
  • Prof. Dr. Bolten, Matthias, Bergische Universität Wuppertal, Germany
organization
publishing date
type
Thesis
publication status
published
subject
keywords
Discontinuous Galerkin Method, Spectral Element Method, Finite Volume Method, Implicit Schemes, Preconditioner, Multigrid Method, Space-Time, Lobatto IIIC Method, Local Fourier Analysis
pages
227 pages
publisher
Lund University
defense location
Hörmander lecture hall (MH), Matematikcentrum, Sölvegatan 18A, Lund. Join via zoom: https://lu-se.zoom.us/j/64731468039pwd=VmRiK01rTmNjbFJ5VlFHL2k2NUxrQT09
defense date
2022-02-28 15:00:00
ISSN
1404-0034
ISBN
978-91-8039-154-2
978-91-8039-153-5
language
English
LU publication?
yes
id
15cc0819-c839-4727-b82b-10ada2ce39c8
date added to LUP
2022-01-24 09:47:12
date last changed
2022-04-06 18:34:03
@phdthesis{15cc0819-c839-4727-b82b-10ada2ce39c8,
  abstract     = {{In this thesis we study efficient solvers for space-time discontinuous Galerkin spectral element methods (DG-SEM). These discretizations result in fully implicit schemes of variable order in both spatial and temporal directions. The popularity of space-time DG methods has increased in recent years and entropy stable space-time DG-SEM have been constructed for conservation laws, making them interesting for these applications. <br/><br/>The size of the nonlinear system resulting from differential equations discretized with space-time DG-SEM is dependent on the order of the method, and the corresponding Jacobian is of block form with dense blocks. Thus, the problem arises to efficiently solve these huge nonlinear systems with regards to CPU time as well as memory consumption. The lack of good solvers for three-dimensional DG applications has been identified as one of the major obstacles before high order methods can be adapted for industrial applications.<br/><br/>It has been proven that DG-SEM in time and Lobatto IIIC Runge-Kutta methods are equivalent, in that both methods lead to the same discrete solution. This allows to implement space-time DG-SEM in two ways: Either as a full space-time system or by decoupling the temporal elements and using implicit time-stepping with Lobatto IIIC methods. We compare theoretical properties and discuss practical aspects of the respective implementations.<br/><br/>When considering the full space-time system, multigrid can be used as solver. We analyze this solver with the local Fourier analysis, which gives more insight into the efficiency of the space-time multigrid method. <br/><br/>The other option is to decouple the temporal elements and use implicit Runge-Kutta time-stepping methods. We suggest to use Jacobian-free Newton-Krylov (JFNK) solvers since they are advantageous memory-wise. An efficient preconditioner for the Krylov sub-solver is needed to improve the convergence speed. However, we want to avoid constructing or storing the Jacobian, otherwise the favorable memory consumption of the JFNK approach would be obsolete. We present a preconditioner based on an auxiliary first order finite volume replacement operator. Based on the replacement operator we construct an agglomeration multigrid preconditioner with efficient smoothers using pseudo time integrators. Then only the Jacobian of the replacement operator needs to be constructed and the DG method is still Jacobian-free. Numerical experiments for hyperbolic test problems as the advection, advection-diffusion and Euler equations in several dimensions demonstrate the potential of the new approach.}},
  author       = {{Versbach, Lea Miko}},
  isbn         = {{978-91-8039-154-2}},
  issn         = {{1404-0034}},
  keywords     = {{Discontinuous Galerkin Method; Spectral Element Method; Finite Volume Method; Implicit Schemes; Preconditioner; Multigrid Method; Space-Time; Lobatto IIIC Method; Local Fourier Analysis}},
  language     = {{eng}},
  publisher    = {{Lund University}},
  school       = {{Lund University}},
  title        = {{Efficient Solvers for Space-Time Discontinuous Galerkin Spectral Element Methods}},
  url          = {{https://lup.lub.lu.se/search/files/112681229/DissertationLeaVersbach_electronic.pdf}},
  year         = {{2022}},
}