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Linders, Viktor
Publications (10 of 12) Show all publications
Linders, V., Nordström, J. & Frankel, S. H. (2019). Convergence and stability properties of summation-by-parts in time. Linköping: Linköping University Electronic Press
Open this publication in new window or tab >>Convergence and stability properties of summation-by-parts in time
2019 (English)Report (Other academic)
Abstract [en]

We extend the list of stability properties satisfied by Summation-By-Parts (SBP) in time to include strong S-stability, dissipative stability and stiff accuracy. Further, it is shown that SBP in time is B-convergent for strictly contractive non-linear problems and weakly convergent for non-linear problems that are both contractive and dissipative

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2019. p. 16
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2019:4
Keywords
SBP intime; Runge-Kuttamethods; S-stability; Stiffaccuracy; Dissipativestabilty; B-convergence
National Category
Computational Mathematics Mathematics
Identifiers
urn:nbn:se:liu:diva-156229 (URN)
Available from: 2019-04-08 Created: 2019-04-08 Last updated: 2019-04-08Bibliographically approved
Linders, V., Lundquist, T. & Nordström, J. (2018). On the order of Accuracy of Finite Difference Operators on Diagonal Norm Based Summation-by-Parts Form. SIAM Journal on Numerical Analysis, 56(2), 1048-1063
Open this publication in new window or tab >>On the order of Accuracy of Finite Difference Operators on Diagonal Norm Based Summation-by-Parts Form
2018 (English)In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 56, no 2, p. 1048-1063Article in journal (Refereed) Published
Abstract [en]

In this paper we generalize results regarding the order of accuracy of finite difference operators on summation-by-parts (SBP) form, previously known to hold on uniform grids, to grids with arbitrary point distributions near domain boundaries. We give a definite proof that the order of accuracy in the interior of a diagonal norm based SBP operator must be at least twice that of the boundary stencil, irrespective of the grid point distribution near the boundary. Additionally, we prove that if the order of accuracy in the interior is precisely twice that of the boundary, then the diagonal norm defines a quadrature rule of the same order as the interior stencil. Again, this result is independent of the grid point distribution near the domain boundaries.

Place, publisher, year, edition, pages
Society for Industrial and Applied Mathematics, 2018
Keywords
finite dierence schemes, summation-by-parts operators, numerical differentiation, quadrature rules, order of accuracy
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-147643 (URN)10.1137/17M1139333 (DOI)000431189500017 ()
Available from: 2018-05-02 Created: 2018-05-02 Last updated: 2018-05-23
Nordström, J. & Linders, V. (2018). Well-posed and stable transmission problems. Journal of Computational Physics, 364, 95-110
Open this publication in new window or tab >>Well-posed and stable transmission problems
2018 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 364, p. 95-110Article in journal (Refereed) Published
Abstract [en]

We introduce the notion of a transmission problem to describe a general class of problems where different dynamics are coupled in time. Well-posedness and stability are analysed for continuous and discrete problems using both strong and weak formulations, and a general transmission condition is obtained. The theory is applied to the coupling of fluid-acoustic models, multi-grid implementations, adaptive mesh refinements, multi-block formulations and numerical filtering.

Place, publisher, year, edition, pages
Elsevier, 2018
Keywords
Transmission problems; Well-posedness; Stability; Adaptive mesh refinement; Numerical filter; Multi-grid
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-146206 (URN)10.1016/j.jcp.2018.03.003 (DOI)000432481000005 ()
Note

Funding agencies: Swedish Meteorological and Hydrological Institute (SMHI)

Available from: 2018-03-29 Created: 2018-03-29 Last updated: 2018-06-14
Delorme, Y. T., Puri, K., Nordström, J., Linders, V., Dong, S. & Frankel, S. H. (2017). A simple and efficient incompressible Navier-Stokes solver for unsteady complex geometry flows on truncated domains. Computers & Fluids, 150, 84-94
Open this publication in new window or tab >>A simple and efficient incompressible Navier-Stokes solver for unsteady complex geometry flows on truncated domains
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2017 (English)In: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 150, p. 84-94Article in journal (Refereed) Published
Abstract [en]

Incompressible Navier-Stokes solvers based on the projection method often require an expensive numerical solution of a Poisson equation for a pressure-like variable. This often involves linear system solvers based on iterative and multigrid methods which may limit the ability to scale to large numbers of processors. The artificial compressibility method (ACM) [6] introduces a time derivative of the pressure into the incompressible form of the continuity equation creating a coupled closed hyperbolic system that does not require a Poisson equation solution and allows for explicit time-marching and localized stencil numerical methods. Such a scheme should theoretically scale well on large numbers of CPUs, GPU'€™s, or hybrid CPU-GPU architectures. The original ACM was only valid for steady flows and dual-time stepping was often used for time-accurate simulations. Recently, Clausen [7] has proposed the entropically damped artificial compressibility (EDAC) method which is applicable to both steady and unsteady flows without the need for dual-time stepping. The EDAC scheme was successfully tested with both a finite-difference MacCormack'€™s method for the two-dimensional lid driven cavity and periodic double shear layer problem and a finite-element method for flow over a square cylinder, with scaling studies on the latter to large numbers of processors. In this study, we discretize the EDAC formulation with a new optimized high-order centered finite-difference scheme and an explicit fourth-order Runge-€“Kutta method. This is combined with an immersed boundary method to efficiently treat complex geometries and a new robust outflow boundary condition to enable higher Reynolds number simulations on truncated domains. Validation studies for the Taylor-€“Green Vortex problem and the lid driven cavity problem in both 2D and 3D are presented. An eddy viscosity subgrid-scale model is used to enable large eddy simulations for the 3D cases. Finally, an application to flow over a sphere is presented to highlight the boundary condition and performance comparisons to a traditional incompressible Navier-€“Stokes solver is shown for the 3D lid driven cavity. Overall, the combined EDAC formulation and discretization is shown to be both effective and affordable.

Place, publisher, year, edition, pages
Elsevier, 2017
Keywords
Artificial compressibility method, EDAC, High-order numerical methods, Large Eddy simulation
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-136507 (URN)10.1016/j.compfluid.2017.03.030 (DOI)000401219000007 ()
Note

Funding agencies: Rosenblatt Chair within the faculty of Mechanical Engineering; Zeff Fellowship Trust

Available from: 2017-04-19 Created: 2017-04-19 Last updated: 2017-11-20Bibliographically approved
Linders, V., Lundquist, T. & Nordström, J. (2017). On the order of Accuracy of Finite Difference Operators on Diagonal Norm Based Summation-By-Parts Form. Linköping: Linköping University Electronic Press
Open this publication in new window or tab >>On the order of Accuracy of Finite Difference Operators on Diagonal Norm Based Summation-By-Parts Form
2017 (English)Report (Other academic)
Abstract [en]

In this paper we generalise results regarding the order of accuracy of finite difference operators on Summation-By-Parts (SBP) form, previously known to hold on uniform grids, to grids with arbitrary point distributions near domain boundaries. We give a definite proof that the order of accuracy in the interior of a diagonal norm based SBP operator must be at least twice that of the boundary stencil, irrespective of the grid point distribution near the boundary. Additionally, we prove that if the order of accuracy in the interior is precisely twice that of the boundary, then the diagonal norm defines a quadrature rule of the same order as the interior stencil. Again, this result is independent of the grid point distribution near the domain boundaries.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2017. p. 15
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2017:11
Keywords
Finite difference schemes, summation-by-parts operators, numerical differentiation, quadrature rules, order of accuracy
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-140815 (URN)
Available from: 2017-09-12 Created: 2017-09-12 Last updated: 2017-11-20Bibliographically approved
Linders, V., Kupiainen, M. & Nordström, J. (2017). Summation-by-Parts Operators with Minimal Dispersion Error for Coarse Grid Flow Calculations. Journal of Computational Physics, 340, 160-176
Open this publication in new window or tab >>Summation-by-Parts Operators with Minimal Dispersion Error for Coarse Grid Flow Calculations
2017 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 340, p. 34p. 160-176Article in journal (Refereed) Published
Abstract [en]

We present a procedure for constructing Summation-by-Parts operators with minimal dispersion error both near and far from numerical interfaces. Examples of such operators are constructed and compared with a higher order non-optimised Summation-by-Parts operator. Experiments show that the optimised operators are superior for wave propagation and turbulent flows involving large wavenumbers, long solution times and large ranges of resolution scales.

Place, publisher, year, edition, pages
Elsevier, 2017. p. 34
Keywords
Summation-by-Parts, Dispersion relation, Finite differences, Wave Propagation
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-136474 (URN)10.1016/j.jcp.2017.03.039 (DOI)000401137900009 ()
Available from: 2017-04-12 Created: 2017-04-12 Last updated: 2017-11-20Bibliographically approved
Nordström, J. & Linders, V. (2017). Well-posed and Stable Transmission Problems. Linköping: Linköping University Electronic Press
Open this publication in new window or tab >>Well-posed and Stable Transmission Problems
2017 (English)Report (Other academic)
Abstract [en]

We introduce the notion of a transmission problem to describe a general class of problems where different dynamics are coupled in time. Well-posedness and stability is analysed for continuous and discrete problems using both strong and weak formulations, and a general transmission condition is obtained. The theory is applied to several examples including the coupling of fluid flow models, multi-grid implementations, multi-block formulations and numerical filtering.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2017. p. 28
Series
LiTH-MAT-R, ISSN 0348-2960 ; 15
Keywords
Initial-boundary value problems, Transmission problems, Energy estimates, Well-posedness, Multi-block, Numerical Filter. Interpolation, Multi-grid, Summation-by-Parts, Stability
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-142348 (URN)LiTH-MAT-R--2017/15--SE (ISRN)
Available from: 2017-10-27 Created: 2017-10-27 Last updated: 2017-11-20Bibliographically approved
Linders, V., Kupiainen, M., Frankel, S. H., Delorme, Y. & Nordström, J. (2016). Summation-by-parts Operators with Minimal Dispersion Error for Accurate and Efficient Flow Calculations. In: : . Paper presented at 54th AIAA Aerospace Sciences Meeting, AIAA SciTech, (AIAA 2016-1329)..
Open this publication in new window or tab >>Summation-by-parts Operators with Minimal Dispersion Error for Accurate and Efficient Flow Calculations
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2016 (English)Conference paper, Published paper (Refereed)
Abstract [en]

We develop summation-by-parts operators with minimal dispersion errors both near and far from boundaries and interfaces. Such operators are superior to classical stencils for problems involving high frequency waves or multi-frequency solutions over long time intervals with a relatively coarse spatial mesh. This is demonstrated by solving the Taylor-Green vortex flow with optimised and classical operators both in a purely periodic setting as well as in the presence of numerical interfaces.

National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-124210 (URN)10.2514/6.2016-1329 (DOI)
Conference
54th AIAA Aerospace Sciences Meeting, AIAA SciTech, (AIAA 2016-1329).
Available from: 2016-01-22 Created: 2016-01-22 Last updated: 2016-01-22
Linders, V., Kupiainen, M. & Nordström, J. (2016). Summation-by-Parts Operators with Minimal Dispersion Error for Coarse Grid Flow Calculations. Linköping: Linköping University Electronic Press
Open this publication in new window or tab >>Summation-by-Parts Operators with Minimal Dispersion Error for Coarse Grid Flow Calculations
2016 (English)Report (Other academic)
Abstract [en]

We present a procedure for constructing Summation-by-Parts operators with minimal dispersion error both near and far from numerical interfaces. Examples of such operators are constructed and compared with purely periodic stencils as well as non-optimised Summation-by-Parts operators of higher order. Experiments show that the optimised operators are superior for wave propagation and turbulent flows involving large wavenumbers, long solution times and large ranges of resolution scales.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2016. p. 34
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2016:7
Keywords
Summation-by-Parts; Dispersion relation; Finite differences; Wave propagation
National Category
Computational Mathematics Mathematics
Identifiers
urn:nbn:se:liu:diva-129447 (URN)
Available from: 2016-06-20 Created: 2016-06-20 Last updated: 2016-09-28Bibliographically approved
Linders, V. & Nordström, J. (2015). Uniformly Best Wavenumber Approximations by Spatial Central Difference Operators. Linköping University Electronic Press
Open this publication in new window or tab >>Uniformly Best Wavenumber Approximations by Spatial Central Difference Operators
2015 (English)Report (Other academic)
Abstract [en]

We construct accurate central difference stencils for problems involving high frequency waves or multi-frequency solutions over long time intervals with a relatively coarse spatial mesh, and with an easily obtained bound on the dispersion error. This is done by demonstrating that the problem of constructing central difference stencils that have minimal dispersion error in the infinity norm can be recast into a problem of approximating a continuous function from a finite dimensional subspace with a basis forming a Chebyshev set. In this new formulation, characterising and numerically obtaining optimised schemes can be done using established theory.

Place, publisher, year, edition, pages
Linköping University Electronic Press, 2015. p. 29
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2014:17
Keywords
Dispersion relation; Wave propagation; Wavenumber approximation; Finite differences; Approximation theory
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-114132 (URN)LiTH-MAT-R--2014/17--SE (ISRN)
Available from: 2015-02-10 Created: 2015-02-10 Last updated: 2015-02-10Bibliographically approved
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