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A new multigrid formulation for high order finite difference methods on summation-by-parts form
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0002-5555-9544
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering. Saab Aerospace, SE-581 88 Linköping, Sweden.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0002-7972-6183
2018 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 359, p. 216-238Article in journal (Refereed) Published
Abstract [en]

Multigrid schemes for high order finite difference methods on summation-by-parts form are studied by comparing the effect of different interpolation operators. By using the standard linear prolongation and restriction operators, the Galerkin condition leads to inaccurate coarse grid discretizations. In this paper, an alternative class of interpolation operators that bypass this issue and preserve the summation-by-parts property on each grid level is considered. Clear improvements of the convergence rate for relevant model problems are achieved.

Place, publisher, year, edition, pages
2018. Vol. 359, p. 216-238
Keywords [en]
High order finite difference methodsSummation-by-partsMultigridRestriction and prolongation operatorsConvergence acceleration
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-145086DOI: 10.1016/j.jcp.2018.01.011ISI: 000427396200011OAI: oai:DiVA.org:liu-145086DiVA, id: diva2:1181770
Note

Funding agencies:  VINNOVA, the Swedish Governmental Agency for Innovation Systems [2013-01209]

Available from: 2018-02-09 Created: 2018-02-09 Last updated: 2019-09-03
In thesis
1. Eigenvalue analysis and convergence acceleration techniques for summation-by-parts approximations
Open this publication in new window or tab >>Eigenvalue analysis and convergence acceleration techniques for summation-by-parts approximations
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Many physical phenomena can be described mathematically by means of partial differential equations. These mathematical formulations are said to be well-posed if a unique solution, bounded by the given data, exists. The boundedness of the solution can be established through the so-called energy-method, which leads to an estimate of the solution by means of integration-by-parts. Numerical approximations mimicking integration-by-parts discretely are said to fulfill the Summation-By-Parts (SBP) property. These formulations naturally yield bounded approximate solutions if the boundary conditions are weakly imposed through Simultaneous-Approximation-Terms (SAT). Discrete problems with bounded solutions are said to be energy-stable. Energy-stable and high-order accurate SBP-SAT discretizations for well-posed linear problems were first introduced for centered finite-difference methods. These mathematical formulations, based on boundary conforming grids, allow for an exact mimicking of integration-by-parts. However, other discretizations techniques that do not include one or both boundary nodes, such as pseudo-spectral collocation methods, only fulfill a generalized SBP (GSBP) property but still lead to energy-stable solutions. This thesis consists of two main topics. The first part, which is mostly devoted to theoretical investigations, treats discretizations based on SBP and GSBP operators. A numerical approximation of a conservation law is said to be conservative if the approximate solution mimics the physical conservation property. It is shown that conservative and energy-stable spatial discretizations of variable coefficient problems require an exact numerical mimicking of integration-by-parts. We also discuss the invertibility of the algebraic problems arising from (G)SBP-SAT discretizations in time of energy-stable spatial approximations. We prove that pseudo-spectral collocation methods for the time derivative lead to invertible fully-discrete problems. The same result is proved for second-, fourth- and sixth-order accurate finite-difference based time integration methods. Once the invertibility of (G)SBP-SAT discrete formulations is established, we are interested in efficient algorithms for the unique solution of such problems. To this end, the second part of the thesis has a stronger experimental flavour and deals with convergence acceleration techniques for SBP-SAT approximations. First, we consider a modified Dual Time-Stepping (DTS) technique which makes use of two derivatives in pseudo-time. The new DTS formulation, compared to the classical one, accelerates the convergence to steady-state and reduces the stiffness of the problem. Next, we investigate multi-grid methods. For parabolic problems, highly oscillating error modes are optimally damped by iterative methods, while smooth residuals are transferred to coarser grids. In this case, we show that the Galerkin condition in combination with the SBP-preserving interpolation operators leads to fast convergence. For hyperbolic problems, low frequency error modes are rapidly expelled by grid coarsening, since coarser grids have milder stability restrictions on time steps. For such problems, Total Variation Dimishing Multi-Grid (TVD-MG) allows for faster wave propagation of first order upwind discretizations. In this thesis, we extend low order TVD-MG schemes to high-order SBP-SAT upwind discretizations.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2019. p. 205
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 2002
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-160073 (URN)978-91-7685-023-7 (ISBN)
Public defence
2019-10-25, Ada Lovelace, Hus B, Campus Valla, Linköping, 13:15 (English)
Opponent
Supervisors
Funder
Vinnova, 2013-01209
Available from: 2019-09-05 Created: 2019-09-03 Last updated: 2019-09-16Bibliographically approved

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The full text will be freely available from 2020-01-12 15:01
Available from 2020-01-12 15:01

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