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Eigenvalue analysis for summation-by-parts finite difference time discretizations
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.ORCID iD: 0000-0002-7972-6183
2019 (English)Report (Other academic)
Abstract [en]

Diagonal norm finite-difference based time integration methods in summation-by-parts form are investigated. The second, fourth and sixth order accurate discretizations are proven to have eigenvalues with strictly positive real parts. This leads to provably invertible fully-discrete approximations of initial boundary value problems.

Our findings also allow us to conclude that the second, fourth and sixth order time discretizations are stiffly accurate, strongly S-stable and dissipatively stable Runge-Kutta methods. The procedure outlined in this article can be extended to even higher order summation-by-parts approximations with repeating stencil.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2019. , p. 35
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2019:9
Keywords [en]
Time integration, Initial value problem, Summation-by-parts operators, Finite difference methods, Eigenvalue problem
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-160009ISRN: LiTH-MAT-R-2019/09-SEOAI: oai:DiVA.org:liu-160009DiVA, id: diva2:1347842
Available from: 2019-09-02 Created: 2019-09-02 Last updated: 2019-09-03Bibliographically approved
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. 38
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 2002
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-160073 (URN)10.3384/diss.diva-160073 (DOI)9789176850237 (ISBN)
Public defence
2019-10-25, Ada Lovelace, B Building, Campus Valla, Linköping, 13:15 (English)
Opponent
Supervisors
Funder
Vinnova, 2013-01209
Available from: 2019-09-05 Created: 2019-09-03 Last updated: 2019-09-23Bibliographically approved

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Ruggiu, Andrea AlessandroNordström, Jan

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