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Ålund, O. & Nordström, J. (2018). A Stable Domain Decomposition Technique for Advection–Diffusion Problems. Journal of Scientific Computing, 1-20
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, p. 1-20Article in journal (Refereed) Epub ahead of print
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)
Available from: 2018-05-14 Created: 2018-05-14 Last updated: 2018-05-14
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
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
Wahlsten, M. & Nordström, J. (2018). Robust boundary conditions for stochastic incompletely parabolic systems of equations. Journal of Computational Physics, 371, 192-213
Open this publication in new window or tab >>Robust boundary conditions for stochastic incompletely parabolic systems of equations
2018 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 371, p. 192-213Article in journal (Refereed) Epub ahead of print
Abstract [en]

We study an incompletely parabolic system in three space dimensions with stochastic boundary and initial data. We show how the variance of the solution can be manipulated by the boundary conditions, while keeping the mean value of the solution unaffected. Estimates of the variance of the solution is presented both analytically and numerically. We exemplify the technique by applying it to an incompletely parabolic model problem, as well as the one-dimensional compressible Navier–Stokes equations.

Keywords
Uncertainty quantification; Incompletely parabolic system; Initial boundary value problems; Stochastic data; Variance reduction; Robust design
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-148214 (URN)10.1016/j.jcp.2018.04.060 (DOI)
Available from: 2018-06-03 Created: 2018-06-03 Last updated: 2018-06-03
Wahlsten, M. & Nordström, J. (2018). Stochastic Galerkin Projection and Numerical Integration for Stochastic Investigations of the Viscous Burgers’ Equation. Linköping: Linköping University Electronic Press
Open this publication in new window or tab >>Stochastic Galerkin Projection and Numerical Integration for Stochastic Investigations of the Viscous Burgers’ Equation
2018 (English)Report (Other academic)
Abstract [en]

We consider a stochastic analysis of the non-linear viscous Burgers’ equation and focus on the comparison between intrusive and non-intrusive uncer- tainty quantification methods. 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. The two methods are applied to a provably stable formulation of the viscous Burgers’ equation, and compared. As measures of comparison: variance size, computational efficiency and accuracy are used.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2018. p. 14
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2018:2
Keywords
Uncertainty quantification, stochastic data, polynomial chaos, stochastic Galerkin, intrusive methods, non-intrusive methods, Burgers’ equation
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-146038 (URN)LiTH-MAT-R--2018/02--SE (ISRN)
Available from: 2018-03-22 Created: 2018-03-22 Last updated: 2018-04-09Bibliographically approved
Nikkar, S. & Nordström, J. (2018). Summation-by-Parts Operators for Non-Simply Connected Domains. SIAM Journal on Scientific Computing, 40(3), 1250-1273
Open this publication in new window or tab >>Summation-by-Parts Operators for Non-Simply Connected Domains
2018 (English)In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 40, no 3, p. 1250-1273Article in journal (Refereed) Published
Abstract [en]

We construct fully discrete stable and accurate numerical schemes for solving partial differential equations posed on non-simply connected spatial domains. The schemes are constructed using summation-by-parts operators in combination with a weak imposition of initial and boundary conditions using the simultaneous approximation term technique. In the theoretical part, we consider the two-dimensional constant coefficient advection equation posed on a rectangular spatial domain with a hole. We construct the new scheme and study well-posedness and stability. Once the theoretical development is done, the technique is extended to more complex non-simply connected geometries. Numerical experiments corroborate the theoretical results and show the applicability of the new approach and its advantages over the standard multiblock technique. Finally, an application using the linearized Euler equations for sound propagation is presented.

Place, publisher, year, edition, pages
Society for Industrial and Applied Mathematics, 2018
Keywords
initial boundary value problems, stability, well-posedness, boundary conditions, non-simply connected domains, complex geometries
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-147651 (URN)10.1137/18M1163671 (DOI)
Available from: 2018-05-03 Created: 2018-05-03 Last updated: 2018-05-03
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
Ruggiu, A. A., Weinerfelt, P. & Nordström, J. (2017). A new multigrid formulation for high order finite difference methods on summation-by-parts form. Linköping: Linköping University Electronic Press
Open this publication in new window or tab >>A new multigrid formulation for high order finite difference methods on summation-by-parts form
2017 (English)Report (Other academic)
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.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2017. p. 32
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2017:08
Keywords
High order finite difference methods, summation-by-parts, multigrid, restriction and prolongation operators, convergence acceleration
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-138955 (URN)LiTH-MAT-R--2017/08--SE (ISRN)
Available from: 2017-06-27 Created: 2017-06-27 Last updated: 2018-01-19Bibliographically approved
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Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0002-7972-6183

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