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Eriksson, S. (2018). A Dual Consistent Finite Difference Method with Narrow Stencil Second Derivative Operators. Journal of Scientific Computing, 75(2), 906-940
Open this publication in new window or tab >>A Dual Consistent Finite Difference Method with Narrow Stencil Second Derivative Operators
2018 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 75, no 2, p. 906-940Article in journal (Refereed) Published
Abstract [en]

We study the numerical solutions of time-dependent systems of partial differential equations, focusing on the implementation of boundary conditions. The numerical method considered is a finite difference scheme constructed by high order summation by parts operators, combined with a boundary procedure using penalties (SBP-SAT). Recently it was shown that SBP-SAT finite difference methods can yield superconvergent functional output if the boundary conditions are imposed such that the discretization is dual consistent. We generalize these results so that they include a broader range of boundary conditions and penalty parameters. The results are also generalized to hold for narrow-stencil second derivative operators. The derivations are supported by numerical experiments.

National Category
Computational Mathematics
Research subject
Mathematics, Applied Mathematics
Identifiers
urn:nbn:se:lnu:diva-77793 (URN)10.1007/s10915-017-0569-6 (DOI)
Available from: 2018-09-14 Created: 2018-09-14 Last updated: 2018-10-09Bibliographically approved
Eriksson, S. & Nordström, J. (2018). Finite difference schemes with transferable interfaces for parabolic problems. Linköping: Linköping University Electronic Press
Open this publication in new window or tab >>Finite difference schemes with transferable interfaces for parabolic problems
2018 (English)Report (Other academic)
Abstract [en]

We derive a method to locally change the order of accuracy of finite difference schemes that approximate the second derivative. The derivation is based on summation-by-parts operators, which are connected at interfaces using penalty terms. At such interfaces, the numerical solution has a double representation, with one representation in each domain. We merge this double representation into a single one, yielding a new scheme with unique solution values in all grid points. The resulting scheme is proven to be stable, accurate and dual consistent.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2018. p. 16
Series
LiTH-MAT-R, ISSN 0348-2960
Keywords
Finite difference methods, summation-by-parts, high order accuracy, dual consistency, superconvergence, interfaces
National Category
Computational Mathematics
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-77790 (URN)
Available from: 2018-09-14 Created: 2018-09-14 Last updated: 2018-10-19Bibliographically approved
Eriksson, S. & Nordström, J. (2018). Finite difference schemes with transferable interfaces for parabolic problems. Journal of Computational Physics, 375, 935-949
Open this publication in new window or tab >>Finite difference schemes with transferable interfaces for parabolic problems
2018 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 375, p. 935-949Article in journal (Refereed) Published
Abstract [en]

We derive a method to locally change the order of accuracy of finite difference schemes that approximate the second derivative. The derivation is based on summation-by-parts operators, which are connected at interfaces using penalty terms. At such interfaces, the numerical solution has a double representation, with one representation in each domain. We merge this double representation into a single one, yielding a new scheme with unique solution values in all grid points. The resulting scheme is proven to be stable, accurate and dual consistent. (C) 2018 Elsevier Inc. All rights reserved.

Place, publisher, year, edition, pages
Elsevier, 2018
Keywords
Finite difference methods, Summation-by-parts, High order accuracy, Dual consistency, Superconvergence, Interfaces
National Category
Mathematics
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-79090 (URN)10.1016/j.jcp.2018.08.051 (DOI)000450907600043 ()2-s2.0-85053206186 (Scopus ID)
Available from: 2018-12-06 Created: 2018-12-06 Last updated: 2019-08-29Bibliographically approved
Eriksson, S. & Nordström, J. (2017). Exact Non-reflecting Boundary Conditions Revisited: Well-Posedness and Stability. Foundations of Computational Mathematics, 17(4), 957-986
Open this publication in new window or tab >>Exact Non-reflecting Boundary Conditions Revisited: Well-Posedness and Stability
2017 (English)In: Foundations of Computational Mathematics, ISSN 1615-3375, E-ISSN 1615-3383, Vol. 17, no 4, p. 957-986Article in journal (Refereed) Published
Abstract [en]

Exact non-reflecting boundary conditions for a linear incompletely parabolic system in one dimension have been studied. The system is a model for the linearized compressible Navier-Stokes equations, but is less complicated which allows for a detailed analysis without approximations. It is shown that well-posedness is a fundamental property of the exact non-reflecting boundary conditions. By using summation by parts operators for the numerical approximation and a weak boundary implementation, it is also shown that energy stability follows automatically.

Place, publisher, year, edition, pages
Springer, 2017
Keywords
Non-reflecting boundary conditions, Well-posedness, Summation by parts, Weak boundary implementation, Stability
National Category
Computational Mathematics
Research subject
Mathematics, Mathematics
Identifiers
urn:nbn:se:lnu:diva-77789 (URN)10.1007/s10208-016-9310-3 (DOI)000407126400004 ()
Available from: 2018-09-14 Created: 2018-09-14 Last updated: 2018-09-18
Eriksson, S. & Nordström, J. (2013). Well-posedness and stability of exact non-reflecting boundary conditions. In: 21st AIAA Computational Fluid Dynamics Conference, Fluid Dynamics and Co-located Conferences: . Paper presented at 21st AIAA Computational Fluid Dynamics Conference, 24-27 June, 2013, San Diego, CA. American Institute of Aeronautics and Astronautics, Article ID 2960.
Open this publication in new window or tab >>Well-posedness and stability of exact non-reflecting boundary conditions
2013 (English)In: 21st AIAA Computational Fluid Dynamics Conference, Fluid Dynamics and Co-located Conferences, American Institute of Aeronautics and Astronautics, 2013, article id 2960Conference paper, Published paper (Refereed)
Place, publisher, year, edition, pages
American Institute of Aeronautics and Astronautics, 2013
Series
Conference Proceeding Series ; 2013-2960
National Category
Computational Mathematics
Research subject
Mathematics, Applied Mathematics
Identifiers
urn:nbn:se:lnu:diva-77775 (URN)10.2514/6.2013-2960 (DOI)
Conference
21st AIAA Computational Fluid Dynamics Conference, 24-27 June, 2013, San Diego, CA
Available from: 2018-09-14 Created: 2018-09-14 Last updated: 2018-10-10Bibliographically approved
Eriksson, S. & Nordström, J. (2012). Exact non-reflecting boundary conditions revisited: well-posedness and stability. Uppsala University
Open this publication in new window or tab >>Exact non-reflecting boundary conditions revisited: well-posedness and stability
2012 (English)Report (Other academic)
Abstract [en]

Exact non-reflecting boundary conditions for an incompletely parabolic system have been studied. It is shown that well-posedness is a fundamental property of the non-reflecting boundary conditions. By using summation by parts operators for the numerical approximation and a weak boundary implementation, energy stability follows automatically. The stability in combination with the high order accuracy results in a reliable, efficient and accurate method. The theory is supported by numerical simulations.

Place, publisher, year, edition, pages
Uppsala University, 2012. p. 31
Series
Technical Report ; 2012-032
National Category
Computational Mathematics
Research subject
Mathematics, Applied Mathematics
Identifiers
urn:nbn:se:lnu:diva-77780 (URN)
Available from: 2018-09-14 Created: 2018-09-14 Last updated: 2019-08-30Bibliographically approved
Eriksson, S. (2012). Stable Numerical Methods with Boundary and Interface Treatment for Applications in Aerodynamics. (Doctoral dissertation). Uppsala: Uppsala University
Open this publication in new window or tab >>Stable Numerical Methods with Boundary and Interface Treatment for Applications in Aerodynamics
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In numerical simulations, problems stemming from aerodynamics pose many challenges for the method used. Some of these are addressed in this thesis, such as the fluid interacting with objects, the presence of shocks, and various types of boundary conditions.

Scenarios of the kind mentioned above are described mathematically by initial boundary value problems (IBVPs). We discretize the IBVPs using high order accurate finite difference schemes on summation by parts form (SBP), combined with weakly imposed boundary conditions, a technique called simultaneous approximation term (SAT). By using the energy method, stability can be shown.

The weak implementation is compared to the more commonly used strong implementation, and it is shown that the weak technique enhances the rate of convergence to steady state for problems with solid wall boundary conditions. The analysis is carried out for a linear problem and supported numerically by simulations of the fully non-linear Navier–Stokes equations.

Another aspect of the boundary treatment is observed for fluid structure interaction problems. When exposed to eigenfrequencies, the coupled system starts oscillating, a phenomenon called flutter. We show that the strong implementation sometimes cause instabilities that can be mistaken for flutter.

Most numerical schemes dealing with flows including shocks are first order accurate to avoid spurious oscillations in the solution. By modifying the SBP-SAT technique, a conservative and energy stable scheme is derived where the order of accuracy can be lowered locally. The new scheme is coupled to a shock-capturing scheme and it retains the high accuracy in smooth regions.

For problems with complicated geometry, one strategy is to couple the finite difference method to the finite volume method. We analyze the accuracy of the latter on unstructured grids. For grids of bad quality the truncation error can be of zeroth order, indicating that the method is inconsistent, but we show that some of the accuracy is recovered.

We also consider artificial boundary closures on unbounded domains. Non-reflecting boundary conditions for an incompletely parabolic problem are derived, and it is shown that they yield well-posedness. The SBP-SAT methodology is employed, and we prove that the discretized problem is stable.

Place, publisher, year, edition, pages
Uppsala: Uppsala University, 2012. p. 26
Series
Acta Universitatis Upsaliensis : Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 985
Keywords
summation by parts, simultaneous approximation term, accuracy, stability, finite difference methods
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
urn:nbn:se:lnu:diva-77769 (URN)978-91-554-8509-2 (ISBN)
Public defence
2012-12-07, Room 2446, Polacksbacken, Lägerhyddsvägen 2D, 10:15 (English)
Opponent
Supervisors
Available from: 2018-09-18 Created: 2018-09-14 Last updated: 2018-10-23Bibliographically approved
Nordström, J., Eriksson, S. & Eliasson, P. (2012). Weak and strong wall boundary procedures and convergence to steady-state of the Navier-Stokes equations. Journal of Computational Physics, 231(14), 4867-4884
Open this publication in new window or tab >>Weak and strong wall boundary procedures and convergence to steady-state of the Navier-Stokes equations
2012 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 231, no 14, p. 4867-4884Article in journal (Refereed) Published
Abstract [en]

We study the influence of different implementations of no-slip solid wall boundary conditions on the convergence to steady-state of the Navier–Stokes equations. The various approaches are investigated using the energy method and an eigenvalue analysis. It is shown that the weak implementation is superior and enhances the convergence to steady-state for coarse meshes. It is also demonstrated that all the stable approaches produce the same convergence rate as the mesh size goes to zero. The numerical results obtained by using a fully nonlinear finite volume solver support the theoretical findings from the linear analysis.

National Category
Computational Mathematics
Research subject
Mathematics, Applied Mathematics
Identifiers
urn:nbn:se:lnu:diva-77770 (URN)10.1016/j.jcp.2012.04.007 (DOI)000304257600021 ()
Available from: 2018-09-14 Created: 2018-09-14 Last updated: 2018-10-03Bibliographically approved
Eriksson, S., Abbas, Q. & Nordström, J. (2011). A stable and conservative method for locally adapting the design order of finite difference schemes. Journal of Computational Physics, 230(11), 4216-4231
Open this publication in new window or tab >>A stable and conservative method for locally adapting the design order of finite difference schemes
2011 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 230, no 11, p. 4216-4231Article in journal (Refereed) Published
Abstract [en]

A procedure to locally change the order of accuracy of finite difference schemes is developed. The development is based on existing Summation-By-Parts operators and a weak interface treatment. The resulting scheme is proven to be accurate and stable.

Numerical experiments verify the theoretical accuracy for smooth solutions. In addition, shock calculations are performed, using a scheme where the developed switching procedure is combined with the MUSCL technique.

National Category
Computational Mathematics Computer Sciences
Research subject
Mathematics, Applied Mathematics
Identifiers
urn:nbn:se:lnu:diva-77771 (URN)10.1016/j.jcp.2010.11.020 (DOI)000290185000007 ()
Available from: 2018-09-14 Created: 2018-09-14 Last updated: 2018-09-25Bibliographically approved
Nordström, J., Eriksson, S. & Eliasson, P. (2011). Weak and Strong Wall Boundary Procedures and Convergence to Steady-State of the Navier-Stokes Equations. Linköping: Linköping University Electronic Press
Open this publication in new window or tab >>Weak and Strong Wall Boundary Procedures and Convergence to Steady-State of the Navier-Stokes Equations
2011 (English)Report (Refereed)
Abstract [en]

We study the influence of different implementations of no-slip solid wall boundary conditions on the convergence to steady-state of the Navier-Stokes equations. The various approaches are investigated using the energy method and an eigenvalue analysis. It is shown that the weak implementation is superior and enhances the convergence to steady-state for coarse meshes. It is also demonstrated that all the stable approaches produce the same convergence rate as the mesh size goes to zero. The numerical results obtained by using a fully nonlinear finite volume solver support the theoretical findings from the linear analysis.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2011. p. 32
Series
LiTH-MAT-R ; 15
Keywords
Navier-Stokes, steady-state, boundary conditions, convergence, summation-by-parts
National Category
Computational Mathematics
Identifiers
urn:nbn:se:lnu:diva-77791 (URN)LiTH-MAT-R--2011/15--SE (ISRN)
Available from: 2018-09-14 Created: 2018-09-14 Last updated: 2018-09-18Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0003-1216-1672

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