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Eigenvalue analysis and convergence acceleration techniques for summation-by-parts approximations
Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.ORCID-id: 0000-0002-5555-9544
2019 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Linköping: Linköping University Electronic Press, 2019. , s. 38
Serie
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 2002
Nationell ämneskategori
Beräkningsmatematik
Identifikatorer
URN: urn:nbn:se:liu:diva-160073DOI: 10.3384/diss.diva-160073ISBN: 9789176850237 (tryckt)OAI: oai:DiVA.org:liu-160073DiVA, id: diva2:1348178
Disputation
2019-10-25, Ada Lovelace, B Building, Campus Valla, Linköping, 13:15 (Engelska)
Opponent
Handledare
Forskningsfinansiär
Vinnova, 2013-01209Tillgänglig från: 2019-09-05 Skapad: 2019-09-03 Senast uppdaterad: 2019-09-23Bibliografiskt granskad
Delarbeten
1. On conservation and stability properties for summation-by-parts schemes
Öppna denna publikation i ny flik eller fönster >>On conservation and stability properties for summation-by-parts schemes
2017 (Engelska)Ingår i: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 344, s. 14s. 451-464Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

We discuss conservative and stable numerical approximations in summation-by-parts form for linear hyperbolic problems with variable coefficients. An extended setting, where the boundary or interface may or may not be included in the grid, is considered. We prove that conservative and stable formulations for variable coefficient problems require a boundary and interface conforming grid and exact numerical mimicking of integration-by-parts. Finally, we comment on how the conclusions from the linear analysis carry over to the nonlinear setting.

Förlag
s. 14
Nyckelord
Hyperbolic problems Summation-by-parts Boundary conditions Interface conditions Stability Conservation
Nationell ämneskategori
Beräkningsmatematik
Identifikatorer
urn:nbn:se:liu:diva-137544 (URN)10.1016/j.jcp.2017.05.002 (DOI)000402481300023 ()
Anmärkning

Funding agencies: VINNOVA [2013-01209]

Tillgänglig från: 2017-05-21 Skapad: 2017-05-21 Senast uppdaterad: 2019-09-03
2. On pseudo-spectral time discretizations in summation-by-parts form
Öppna denna publikation i ny flik eller fönster >>On pseudo-spectral time discretizations in summation-by-parts form
2018 (Engelska)Ingår i: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 360, s. 192-201Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

Fully-implicit discrete formulations in summation-by-parts form for initial-boundary value problems must be invertible in order to provide well functioning procedures. We prove that, under mild assumptions, pseudo-spectral collocation methods for the time derivative lead to invertible discrete systems when energy-stable spatial discretizations are used.

Ort, förlag, år, upplaga, sidor
Springer Publishing Company, 2018
Nyckelord
Time integration; Initial boundary value problem; Summation-by-parts operators; Pseudo-spectral methods; Eigenvalue problem
Nationell ämneskategori
Matematik
Identifikatorer
urn:nbn:se:liu:diva-145083 (URN)10.1016/j.jcp.2018.01.043 (DOI)000428966300011 ()2-s2.0-85041575964 (Scopus ID)
Tillgänglig från: 2018-02-09 Skapad: 2018-02-09 Senast uppdaterad: 2019-09-03Bibliografiskt granskad
3. Eigenvalue analysis for summation-by-parts finite difference time discretizations
Öppna denna publikation i ny flik eller fönster >>Eigenvalue analysis for summation-by-parts finite difference time discretizations
2019 (Engelska)Rapport (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Linköping: Linköping University Electronic Press, 2019. s. 35
Serie
LiTH-MAT-R, ISSN 0348-2960 ; 2019:9
Nyckelord
Time integration, Initial value problem, Summation-by-parts operators, Finite difference methods, Eigenvalue problem
Nationell ämneskategori
Matematik
Identifikatorer
urn:nbn:se:liu:diva-160009 (URN)LiTH-MAT-R-2019/09-SE (ISRN)
Tillgänglig från: 2019-09-02 Skapad: 2019-09-02 Senast uppdaterad: 2019-09-03Bibliografiskt granskad
4. Dual Time-Stepping Using Second Derivatives
Öppna denna publikation i ny flik eller fönster >>Dual Time-Stepping Using Second Derivatives
2019 (Engelska)Ingår i: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 81, nr 2, s. 1050-1071Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

We present a modified formulation of the dual time-stepping technique which makes use of two derivatives in pseudo-time. This new technique retains and improves the convergence properties to the stationary solution. When compared with the conventional dual time-stepping, the method with two derivatives reduces the stiffness of the problem and requires fewer iterations for full convergence to steady-state. In the current formulation, these positive effects require that an approximation of the square root of the spatial operator is available and inexpensive.

Ort, förlag, år, upplaga, sidor
Springer, 2019
Nationell ämneskategori
Matematik
Identifikatorer
urn:nbn:se:liu:diva-160245 (URN)10.1007/s10915-019-01047-5 (DOI)000491440200017 ()
Anmärkning

Funding agencies:  Linkoping University; Swedish Governmental Agency for Innovation SystemsVinnova [2013-01209]; VINNOVAVinnova

Tillgänglig från: 2019-09-13 Skapad: 2019-09-13 Senast uppdaterad: 2019-11-05
5. A new multigrid formulation for high order finite difference methods on summation-by-parts form
Öppna denna publikation i ny flik eller fönster >>A new multigrid formulation for high order finite difference methods on summation-by-parts form
2018 (Engelska)Ingår i: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 359, s. 216-238Artikel i tidskrift (Refereegranskat) 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.

Nyckelord
High order finite difference methodsSummation-by-partsMultigridRestriction and prolongation operatorsConvergence acceleration
Nationell ämneskategori
Matematik
Identifikatorer
urn:nbn:se:liu:diva-145086 (URN)10.1016/j.jcp.2018.01.011 (DOI)000427396200011 ()
Anmärkning

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

Tillgänglig från: 2018-02-09 Skapad: 2018-02-09 Senast uppdaterad: 2019-09-03
6. Multigrid schemes for high order discretizations of hyperbolic problems
Öppna denna publikation i ny flik eller fönster >>Multigrid schemes for high order discretizations of hyperbolic problems
2019 (Engelska)Ingår i: 2019 AIAA Aerospace Sciences Meeting, AIAA Scitech Forum, American Institute of Aeronautics and Astronautics, 2019, s. 1-25, artikel-id AIAA 2019-0103Konferensbidrag, Publicerat paper (Refereegranskat)
Abstract [en]

Total variation diminishing multigrid methods have been developed for first order accurate discretizations of hyperbolic conservation laws. This technique is based on a so-called upwind biased residual interpolation and allows for algorithms devoid of spurious numerical oscillations in the transient phase. In this paper, we justify the introduction of such prolongation and restriction operators by rewriting the algorithm in a matrix-vector notation. This perspective sheds new light on multigrid procedures for hyperbolic problems and provides a direct extension for high order accurate difference approximations. The new multigrid procedure is presented, advantages and disadvantages are discussed and numerical experiments are performed.

Ort, förlag, år, upplaga, sidor
American Institute of Aeronautics and Astronautics, 2019
Nationell ämneskategori
Matematik
Identifikatorer
urn:nbn:se:liu:diva-154393 (URN)10.2514/6.2019-0103 (DOI)978-1-62410-578-4 (ISBN)
Konferens
2019 AIAA Aerospace Sciences Meeting, AIAA Scitech Forum, San Diego, California, 7-11 January 2019
Tillgänglig från: 2019-02-11 Skapad: 2019-02-11 Senast uppdaterad: 2019-09-03

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