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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. 32 p.
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2017:08
Keyword
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
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, 84-94 p.Article 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
Keyword
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
Ghasemi, F. & Nordström, J. (2017). Coupling Requirements for Multiphysics Problems Posed on Two Domains. SIAM Journal on Numerical Analysis, 55(6), 2885-2904.
Open this publication in new window or tab >>Coupling Requirements for Multiphysics Problems Posed on Two Domains
2017 (English)In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 55, no 6, 2885-2904 p.Article in journal (Refereed) Published
Abstract [en]

We consider two hyperbolic systems in first order form of different size posed on two domains. Our ambition is to derive general conditions for when the two systems can and cannot be coupled. The adjoint equations are derived and well-posedness of the primal and dual problems is discussed. By applying the energy method, interface conditions for the primal and dual problems are derived such that the continuous problems are well posed. The equations are discretized using a high order finite difference method in summation-by-parts form and the interface conditions are imposed weakly in a stable way, using penalty formulations. It is shown that one specic choice of penalty matrices leads to a dual consistent scheme. By considering an example, it is shown that the correct physical coupling conditions are contained in the set of well posed coupling conditions. It is also shown that dual consistency leads to superconverging functionals and reduced stiffness.

Place, publisher, year, edition, pages
Society for Industrial and Applied Mathematics, 2017
Keyword
well posed problems, high order finite diffrences, stability, summation-by-parts, weak interface conditions, dual consistency, stiffness, superconvergence
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-143261 (URN)10.1137/16M1087710 (DOI)000418663500015 ()
Available from: 2017-11-28 Created: 2017-11-28 Last updated: 2018-01-12Bibliographically approved
O'Reilly, O., Lundquist, T., Dunham, E. M. & Nordström, J. (2017). Energy stable and high-order-accurate finite difference methods on staggered grids. Journal of Computational Physics, 346, 572-589.
Open this publication in new window or tab >>Energy stable and high-order-accurate finite difference methods on staggered grids
2017 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 346, 18 p.572-589 p.Article in journal (Refereed) Published
Abstract [en]

For wave propagation over distances of many wavelengths, high-order finite difference methods on staggered grids are widely used due to their excellent dispersion properties. However, the enforcement of boundary conditions in a stable manner and treatment of interface problems with discontinuous coefficients usually pose many challenges. In this work, we construct a provably stable and high-order-accurate finite difference method on staggered grids that can be applied to a broad class of boundary and interface problems. The staggered grid difference operators are in summation-by-parts form and when combined with a weak enforcement of the boundary conditions, lead to an energy stable method on multiblock grids. The general applicability of the method is demonstrated by simulating an explosive acoustic source, generating waves reflecting against a free surface and material discontinuity.

Place, publisher, year, edition, pages
Academic Press, 2017. 18 p.
Keyword
Staggered grids High order finite difference methods Summation-by-parts Weakly enforced boundary conditions Energy stability Wave propagation
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-139343 (URN)10.1016/j.jcp.2017.06.030 (DOI)000406465000029 ()2-s2.0-85021952350 (Scopus ID)
Note

Funding agencies: Department of Geophysics at Stanford University

Available from: 2017-07-11 Created: 2017-07-11 Last updated: 2017-08-22Bibliographically approved
Nordström, J. & La Cognata, C. (2017). Energy Stable Boundary Conditions for the Nonlinear Incompressible Navier-Stokes Equations. Linköping University Electronic Press.
Open this publication in new window or tab >>Energy Stable Boundary Conditions for the Nonlinear Incompressible Navier-Stokes Equations
2017 (English)Report (Other academic)
Abstract [en]

The nonlinear incompressible Navier-Stokes equations with 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 difference 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.

Place, publisher, year, edition, pages
Linköping University Electronic Press, 2017. 31 p.
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2017:9
Keyword
Navier-Stokes equations, incompressible, boundary conditions, energy estimate, stability, summation-by-parts, high-order accuracy, divergence free
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-139730 (URN)LiTH-MAT-R--2017/09--SE (ISRN)
Available from: 2017-08-14 Created: 2017-08-14 Last updated: 2017-09-05Bibliographically 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, 957-986 p.Article 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
Keyword
Non-reflecting boundary conditions, Well-posedness, Summation by parts, Weak boundary implementation, Stability
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-127034 (URN)10.1007/s10208-016-9310-3 (DOI)000407126400004 ()
Available from: 2016-04-13 Created: 2016-04-13 Last updated: 2017-08-28Bibliographically 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. 15 p.
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2017:11
Keyword
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
La Cognata, C. & Nordström, J. (2017). Spectral analysis of the incompressible Navier-Stokes equations with different boundary conditions. Linköping: Linköping University Electronic Press.
Open this publication in new window or tab >>Spectral analysis of the incompressible Navier-Stokes equations with different boundary conditions
2017 (English)Report (Other academic)
Abstract [en]

The influence of boundary conditions on the spectrum of the incompressible Navier-Stokes equations is studied. The spectra associated to different types of boundary conditions are derived using the Fourier-Laplace technique. In particular, the effect of various combinations of generalized in- and outgoing variables on the convergence to the steady state is investigated. The boundary conditions are analysed in both the continuous and semi-discrete problems. In the latter, high-order schemes in summation-by-parts form with weakly imposed boundary conditions are used to approximate the equations. Numerical calculations are performed and show that the discrete behaviour agrees with the theoretical analysis.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2017. 34 p.
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2017:10
Keyword
Incompressible flows, Navier-Stokes, Fourier-Laplace technique, summation-by-parts, weak boundary conditions
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-139729 (URN)LiTH-MAT-R--2017/10—SE (ISRN)
Available from: 2017-08-14 Created: 2017-08-14 Last updated: 2017-09-05Bibliographically 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, 34 p.160-176 p.Article 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. 34 p.
Keyword
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. 28 p.
Series
LiTH-MAT-R, ISSN 0348-2960 ; 15
Keyword
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
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0002-7972-6183

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