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Nikkar, S. & Nordström, J. (2019). A dual consistent summation-by-parts formulation for the linearized incompressible Navier-Stokes equations posed on deforming domains. Journal of Computational Physics, 376, 322-338
Open this publication in new window or tab >>A dual consistent summation-by-parts formulation for the linearized incompressible Navier-Stokes equations posed on deforming domains
2019 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 376, p. 26p. 322-338Article in journal (Refereed) Published
Abstract [en]

In this article, well-posedness and dual consistency of the linearized constant coefficient incompressible Navier–Stokes equations posed on time-dependent spatial domains are studied. To simplify the derivation of the dual problem and improve the accuracy of gradients, the second order formulation is transformed to first order form. Boundary conditions that simultaneously lead to boundedness of the primal and dual problems are derived.Fully discrete finite difference schemes on summation-by-parts form, in combination with the simultaneous approximation technique, are constructed. We prove energy stability and discrete dual consistency and show how to construct the penalty operators such that the scheme automatically adjusts to the variations of the spatial domain. As a result of the aforementioned formulations, stability and discrete dual consistency follow simultaneously.The method is illustrated by considering a deforming time-dependent spatial domain in two dimensions. The numerical calculations are performed using high order operators in space and time. The results corroborate the stability of the scheme and the accuracy of the solution. We also show that linear functionals are superconverging. Additionally, we investigate the convergence of non-linear functionals and the divergence of the solution.

Place, publisher, year, edition, pages
Elsevier, 2019. p. 26
Keywords
Incompressible Navier-Stokes equations, Deforming domain, Stability, Dual consistency, High order accuracy, Superconvergence
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-152031 (URN)10.1016/j.jcp.2018.09.006 (DOI)000450337400016 ()2-s2.0-85054431823 (Scopus ID)
Available from: 2018-10-17 Created: 2018-10-17 Last updated: 2018-12-13Bibliographically approved
Ghasemi Zinatabadi, F. & Nordström, J. (2019). An Energy Stable Coupling Procedure for the Compressible and Incompressible Navier-Stokes Equations. Linköping: Linköping University Electronic Press
Open this publication in new window or tab >>An Energy Stable Coupling Procedure for the Compressible and Incompressible Navier-Stokes Equations
2019 (English)Report (Other academic)
Abstract [en]

The coupling of the compressible and incompressible Navier-Stokes equations is considered. Our ambition is to take a first step towards a provably well posed and stable coupling procedure. We study a simplified setting with a stationary planar interface and small disturbances from a steady background ow with zero velocity normal to the interface.

The simplified setting motivates the use of the linearized equations, and we derive interface conditions such that the continuous problem satisfy an energy estimate. The interface conditions can be imposed both strongly and weakly. It is shown that the weak and strong interface imposition produce similar continuous energy estimates.

We discretize the problem in time and space by employing finite difference operators that satisfy a summation-by-parts rule. The interface and initial conditions are imposed weakly using a penalty formulation. It is shown that the results obtained for the weak interface conditions in the continuous case, lead directly to stability of the fully discrete problem.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2019. p. 42
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2019:7
Keywords
Compressible fluid, incompressible fluid, Navier-Stokes equations, energy estimate, interface conditions, stability
National Category
Computational Mathematics Mathematics
Identifiers
urn:nbn:se:liu:diva-158905 (URN)LiTH-MAT-R--2019/07--SE (ISRN)
Available from: 2019-07-17 Created: 2019-07-17 Last updated: 2019-07-29Bibliographically approved
Ghasemi Zinatabadi, F. & Nordström, J. (2019). An Energy Stable Coupling Procedure for the Compressible and Incompressible Navier-Stokes Equations. Linköping: Linköping University Electronic Press
Open this publication in new window or tab >>An Energy Stable Coupling Procedure for the Compressible and Incompressible Navier-Stokes Equations
2019 (English)Report (Other academic)
Abstract [en]

The coupling of the compressible and incompressible Navier-Stokes equations is considered. Our ambition is to take a first step towards a provably well posed and stable coupling procedure. We study a simplified setting with a stationary planar interface and small disturbances from a steady background ow with zero velocity normal to the interface.

The simplified setting motivates the use of the linearized equations, and we derive interface conditions such that the continuous problem satisfy an energy estimate. The interface conditions can be imposed both strongly and weakly. It is shown that the weak and strong interface imposition produce similar continuous energy estimates.

We discretize the problem in time and space by employing finite difference operators that satisfy a summation-by-parts rule. The interface and initial conditions are imposed weakly using a penalty formulation. It is shown that the results obtained for the weak interface conditions in the continuous case, lead directly to stability of the fully discrete problem.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2019. p. 42
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2019:7
Keywords
Compressible uid, incompressible uid, Navier-Stokes equations, energy estimate, interface conditions, stability
National Category
Computational Mathematics Mathematics
Identifiers
urn:nbn:se:liu:diva-159531 (URN)LiTH-MAT-R--2019/07--SE (ISRN)
Available from: 2019-08-12 Created: 2019-08-12 Last updated: 2019-08-12Bibliographically approved
Ghasemi, F. & Nordström, J. (2019). An energy stable coupling procedure for the compressible and incompressible Navier-Stokes equations. Journal of Computational Physics, 396, 280-302
Open this publication in new window or tab >>An energy stable coupling procedure for the compressible and incompressible Navier-Stokes equations
2019 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 396, p. 280-302Article in journal (Refereed) Published
Abstract [en]

The coupling of the compressible and incompressible Navier-Stokes equations is considered. Our ambition is to take a first step towards a provably well posed and stable coupling procedure. We study a simplified setting with a stationary planar interface and small disturbances from a steady background flow with zero velocity normal to the interface. The simplified setting motivates the use of the linearized equations, and we derive interface conditions such that the continuous problem satisfy an energy estimate. The interface conditions can be imposed both strongly and weakly. It is shown that the weak and strong interface imposition produce similar continuous energy estimates. We discretize the problem in time and space by employing finite difference operators that satisfy a summation-by-parts rule. The interface and initial conditions are imposed weakly using a penalty formulation. It is shown that the results obtained for the weak interface conditions in the continuous case, lead directly to stability of the fully discrete problem.

Place, publisher, year, edition, pages
Elsevier, 2019
Keywords
Compressible fluid, Incompressible fluid, Navier-Stokes equations, Energy estimate, Interface conditions, Stability
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-159116 (URN)10.1016/j.jcp.2019.07.022 (DOI)000481732600015 ()
Available from: 2019-07-29 Created: 2019-07-29 Last updated: 2019-09-09
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
Nordström, J. & Ruggiu, A. A. (2019). Dual Time-Stepping Using Second Derivatives. Journal of Scientific Computing
Open this publication in new window or tab >>Dual Time-Stepping Using Second Derivatives
2019 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691Article in journal (Refereed) Epub ahead of print
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.

Place, publisher, year, edition, pages
Springer, 2019
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-160245 (URN)10.1007/s10915-019-01047-5 (DOI)
Available from: 2019-09-13 Created: 2019-09-13 Last updated: 2019-09-13
Nordström, J. & Ruggiu, A. A. (2019). Dual Time-Stepping Using Second Derivatives. Linköping: Linköping University Electronic Press
Open this publication in new window or tab >>Dual Time-Stepping Using Second Derivatives
2019 (English)Report (Other academic)
Abstract [en]

We present a modied 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 stiness of the problem and requires fewer iterations for full convergence to steady-state. In the current formulation, these positive eects require that an approximation of the square root of the spatial operator is available and inexpensive.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2019. p. 30
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2019:10
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-160010 (URN)LiTH-MAT-R-2019/10-SE (ISRN)
Available from: 2019-09-02 Created: 2019-09-02 Last updated: 2019-09-13Bibliographically approved
Ruggiu, A. A. & Nordström, J. (2019). Eigenvalue analysis for summation-by-parts finite difference time discretizations. Linköping: Linköping University Electronic Press
Open this publication in new window or tab >>Eigenvalue analysis for summation-by-parts finite difference time discretizations
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
Time integration, Initial value problem, Summation-by-parts operators, Finite difference methods, Eigenvalue problem
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-160009 (URN)LiTH-MAT-R-2019/09-SE (ISRN)
Available from: 2019-09-02 Created: 2019-09-02 Last updated: 2019-09-03Bibliographically approved
Ålund, O. & Nordström, J. (2019). Encapsulated high order difference operators on curvilinear non-conforming grids. Journal of Computational Physics, 385, 209-224
Open this publication in new window or tab >>Encapsulated high order difference operators on curvilinear non-conforming grids
2019 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 385, p. 209-224Article in journal (Refereed) Published
Abstract [en]

Constructing stable difference schemes on complex geometries is an arduous task. Even fairly simple partial differential equations end up very convoluted in their discretized form, making them difficult to implement and manage. Spatial discretizations using so called summation-by-parts operators have mitigated this issue to some extent, particularly on rectangular domains, making it possible to formulate stable discretizations in a compact and understandable manner. However, the simplicity of these formulations is lost for curvilinear grids, where the standard procedure is to transform the grid to a rectangular one, and change the structure of the original equation. In this paper we reinterpret the grid transformation as a transformation of the summation-by-parts operators. This results in operators acting directly on the curvilinear grid. Together with previous developments in the field of nonconforming grid couplings we can formulate simple, implementable, and provably stable schemes on general nonconforming curvilinear grids. The theory is applicable to methods on summation-by-parts form, including finite differences, discontinuous Galerkin spectral element, finite volume, and flux reconstruction methods. Time dependent advection–diffusion simulations corroborate the theoretical development.

Keywords
Non-conforming grids, Curvilinear mappings, Weak interface couplings, Summation-by-parts, Stability, Energy method
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-154938 (URN)10.1016/j.jcp.2019.02.007 (DOI)000460889200011 ()
Available from: 2019-03-06 Created: 2019-03-06 Last updated: 2019-04-01
Nordström, J. & La Cognata, C. (2019). Energy Stable Boundary Conditions for the Nonlinear Incompressible Navier-Stokes Equations. Mathematics of Computation, 88(316), 665-690
Open this publication in new window or tab >>Energy Stable Boundary Conditions for the Nonlinear Incompressible Navier-Stokes Equations
2019 (English)In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 88, no 316, p. 665-690Article in journal (Refereed) Published
Abstract [en]

The nonlinear incompressible Navier-Stokes equations with different types of boundary conditions at far fields and solid walls is considered. Two different formulations of boundary conditions are derived using the energy method. Both formulations are implemented in both strong and weak form and lead to an estimate of the velocity field.

Equipped with energy bounding boundary conditions, the problem is approximated by using discrete derivative operators on summation-by-parts form and weak boundary and initial conditions. By mimicking the continuous analysis, the resulting semi-discrete as well as fully discrete scheme are shown to be provably stable, divergence free, and high-order accurate.

Keywords
Navier-Stokes equations, incompressible, boundary conditions, energy estimate, stability, summation-by-parts, high-order accuracy, divergence free.
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-153296 (URN)10.1090/mcom/3375 (DOI)000452419800007 ()
Available from: 2018-12-10 Created: 2018-12-10 Last updated: 2018-12-20
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0002-7972-6183

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