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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
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
Nordström, J. & Wahlsten, M. (2019). Robust Design of Initial Boundary Value Problems. In: Hirsch, C.; Wunsch, D.; Szumbarski, J.; Łaniewski-Wołłk, Ł.; Pons-Prats, J. (Ed.), Uncertainty Management for Robust Industrial Design in Aeronautics: Findings and Best Practice Collected During UMRIDA, a Collaborative Research Project (2013–2016) Funded by the European Union (pp. 463-478). Springer
Open this publication in new window or tab >>Robust Design of Initial Boundary Value Problems
2019 (English)In: Uncertainty Management for Robust Industrial Design in Aeronautics: Findings and Best Practice Collected During UMRIDA, a Collaborative Research Project (2013–2016) Funded by the European Union / [ed] Hirsch, C.; Wunsch, D.; Szumbarski, J.; Łaniewski-Wołłk, Ł.; Pons-Prats, J., Springer, 2019, p. 463-478Chapter in book (Refereed)
Abstract [en]

We study hyperbolic and incompletely parabolic systems with stochastic boundary and initial data. Estimates of the variance of the solution are presented both analytically and numerically. It is shown that one can reduce the variance for a given input, with a specific choice of boundary condition. The technique is applied to the Maxwell, Euler, and Navier–Stokes equations. Numerical calculations corroborate the theoretical conclusions.

Place, publisher, year, edition, pages
Springer, 2019
Series
Notes on Numerical Fluid Mechanics and Multidisciplinary Design, ISSN 1612-2909, E-ISSN 1860-0824 ; 140
Keywords
Uncertainty quantification, Hyperbolic systems, Incompletely parabolic systems, Initial boundary value problems, Stochastic data, Variance reduction, Robust design
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-150373 (URN)10.1007/978-3-319-77767-2_29 (DOI)9783319777672 (ISBN)9783319777665 (ISBN)
Available from: 2018-08-21 Created: 2018-08-21 Last updated: 2018-08-21
Ålund, O. & Nordström, J. (2018). A Stable Domain Decomposition Technique for Advection–Diffusion Problems. Journal of Scientific Computing, 77(2), 755-774
Open this publication in new window or tab >>A Stable Domain Decomposition Technique for Advection–Diffusion Problems
2018 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 77, no 2, p. 755-774Article in journal (Refereed) Published
Abstract [en]

The use of implicit methods for numerical time integration typically generates very large systems of equations, often too large to fit in memory. To address this it is necessary to investigate ways to reduce the sizes of the involved linear systems. We describe a domain decomposition approach for the advection–diffusion equation, based on the Summation-by-Parts technique in both time and space. The domain is partitioned into non-overlapping subdomains. A linear system consisting only of interface components is isolated by solving independent subdomain-sized problems. The full solution is then computed in terms of the interface components. The Summation-by-Parts technique provides a solid theoretical framework in which we can mimic the continuous energy method, allowing us to prove both stability and invertibility of the scheme. In a numerical study we show that single-domain implementations of Summation-by-Parts based time integration can be improved upon significantly. Using our proposed method we are able to compute solutions for grid resolutions that cannot be handled efficiently using a single-domain formulation. An order of magnitude speed-up is observed, both compared to a single-domain formulation and to explicit Runge–Kutta time integration.

Keywords
Domain decomposition, Partial differential equations, Summation-by-Parts, Finite difference methods, Stability
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-147768 (URN)10.1007/s10915-018-0722-x (DOI)000446594600003 ()
Available from: 2018-05-14 Created: 2018-05-14 Last updated: 2018-10-17
Wahlsten, M. & Nordström, J. (2018). An efficient hybrid method for uncertainty quantification. Linköping: Linköping University Electronic Press
Open this publication in new window or tab >>An efficient hybrid method for uncertainty quantification
2018 (English)Report (Other academic)
Abstract [en]

A technique for coupling an intrusive and non-intrusive uncertainty quantification method is proposed. The intrusive approach uses a combination of polynomial chaos and stochastic Galerkin projection. The non-intrusive method uses numerical integration by combining quadrature rules and the probability density functions of the prescribed uncertainties. A strongly stable coupling procedure between the two methods at an interface is constructed. The efficiency of the hybrid method is exemplified using a hyperbolic system of equations, and verified by numerical experiments.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2018. p. 20
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2018:3
Keywords
Uncertainty quantification, numerical integration, stochastic Galerkin, polynomial chaos, coupling, projection operator
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-146036 (URN)LiTH-MAT-R--2018/03--SE (ISRN)
Available from: 2018-03-22 Created: 2018-03-22 Last updated: 2018-04-09Bibliographically approved
Nordström, J. & Ghasemi, F. (2018). Corrigendum to “On the relation between conservation and dual consistency for summation-by-parts schemes”[J. Comput. Phys. 344 (2017) 437–439]. Journal of Computational Physics, 360, 247-247
Open this publication in new window or tab >>Corrigendum to “On the relation between conservation and dual consistency for summation-by-parts schemes”[J. Comput. Phys. 344 (2017) 437–439]
2018 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 360, p. 247-247Article in journal (Refereed) Published
Place, publisher, year, edition, pages
Academic Press, 2018
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-145718 (URN)10.1016/j.jcp.2018.02.046 (DOI)000428966300014 ()
Available from: 2018-03-19 Created: 2018-03-19 Last updated: 2018-05-18Bibliographically 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 ; 2018:1
Keywords
Finite difference methods, summation-by-parts, high order accuracy, dual consistency, superconvergence, interfaces
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-146078 (URN)LiTH-MAT-R--2018/01--SE (ISRN)
Available from: 2018-03-26 Created: 2018-03-26 Last updated: 2018-04-06Bibliographically 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.

Place, publisher, year, edition, pages
Elsevier, 2018
Keywords
Finite difference methods, Summation-by-parts, High order accuracy, Dual consistency, Superconvergence, Interfaces
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-151404 (URN)10.1016/j.jcp.2018.08.051 (DOI)000450907600043 ()
Available from: 2018-09-19 Created: 2018-09-19 Last updated: 2018-12-13
Pettersson, P., Doostan, A. & Nordström, J. (2018). Level Set Methods for Stochastic Discontinuity Detection in Nonlinear Problems. Linköping: Linköping University Electronic Press
Open this publication in new window or tab >>Level Set Methods for Stochastic Discontinuity Detection in Nonlinear Problems
2018 (English)Report (Other academic)
Abstract [en]

Stochastic physical problems governed by nonlinear conservation laws are challenging due to solution discontinuities in stochastic and physical space. In this paper, we present a level set method to track discontinuities in stochastic space by solving a Hamilton-Jacobi equation. By introducing a speedfunction that vanishes at discontinuities, the iso-zero of the level set problem coincide with the discontinuities of the conservation law. The level set problem is solved on a sequence of successively finer grids in stochastic space. The method is adaptive in the sense that costly evaluations of the conservation law of interest are only performed in the vicinity of the discontinuities during the renement stage. In regions of stochastic space where the solutionis smooth, a surrogate method replaces expensive evaluations of the conservation law. The proposed method is tested in conjunction with different sets of localized orthogonal basis functions on simplex elements, as well as frames based on piecewise polynomials conforming to the level set function. The performance of the proposed method is compared to existing adaptive multi-element generalized polynomial chaos methods.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2018. p. 38
Keywords
Uncertainty quantication, Discontinuity tracking, Level set methods, Polynomial chaos, Hyperbolic PDEs
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-152033 (URN)LiTH-MAT-R--2018/11--SE (ISRN)
Available from: 2018-10-17 Created: 2018-10-17 Last updated: 2018-10-17
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
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0002-7972-6183

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