165 results on '"Jan Nordström"'
Search Results
2. Summation-by-Parts Operators for General Function Spaces
- Author
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Jan Glaubitz, Jan Nordström, and Philipp Öffner
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mimetic discretization ,Matematik ,Numerical Analysis ,Applied Mathematics ,radial basis functions ,Numerical Analysis (math.NA) ,trigonometric functions ,65M12, 65M60, 65M70, 65D25, 65T40, 65D12 ,Computational Mathematics ,FOS: Mathematics ,Mathematics - Numerical Analysis ,general function spaces ,exponential functions ,summation-by-parts operators ,Mathematics - Abstract
Summation-by-parts (SBP) operators are popular building blocks for systematically developing stable and high-order accurate numerical methods for time-dependent differential equations. The main idea behind existing SBP operators is that the solution is assumed to be well approximated by polynomials up to a certain degree, and the SBP operator should therefore be exact for them. However, polynomials might not provide the best approximation for some problems, and other approximation spaces may be more appropriate. In this paper, a theory for SBP operators based on general function spaces is developed. We demonstrate that most of the established results for polynomial-based SBP operators carry over to this general class of SBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than currently known. We exemplify the general theory by considering trigonometric, exponential, and radial basis functions., 22 pages, 6 figures
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- 2023
3. On the order reduction of approximations of fractional derivatives: an explanation and a cure
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Byron A. Jacobs, Fredrik Laurén, and Jan Nordström
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Computational Mathematics ,Matematik ,Computer Networks and Communications ,Applied Mathematics ,Software ,Mathematics ,Fractional derivative ,High-order numerical approximation ,Finite-differences ,Closed quadrature rules ,Summation-by-parts operators - Abstract
Finite-difference based approaches are common for approximating the Caputo fractional derivative. Often, these methods lead to a reduction of the convergence rate that depends on the fractional order. In this note, we approximate the expressions in the fractional derivative components using a separate quadrature rule for the integral and a separate discretization of the derivative in the integrand. By this approach, the error terms from the Newton–Cotes quadrature and the differentiation are isolated and it is possible to conclude that the order dependent error is inevitable when the end points of the interval are included in the quadrature. Furthermore, we show experimentally that the theoretical findings carries over to quadrature rules without the end points included. Finally we show how to increase accuracy for smooth functions, and compensate for the order dependent loss. Funding: BAJ acknowledges support from the National Research Foundation of South Africa under Grant Numbers 129119 and 127567. FL and JN were supported by Vetenskapsrådet, Sweden Grant Numbers 2018-05084 and 2021-05484.
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- 2023
4. Analysis of the SBP-SAT Stabilization for Finite Element Methods Part II: Entropy Stability
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Rémi Abgrall, Philipp Öffner, Jan Nordström, and Svetlana Tokareva
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Work (thermodynamics) ,Series (mathematics) ,Applied Mathematics ,Boundary (topology) ,010103 numerical & computational mathematics ,01 natural sciences ,Stability (probability) ,Finite element method ,010101 applied mathematics ,Computational Mathematics ,Entropy (classical thermodynamics) ,Nonlinear system ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
In the hyperbolic research community, there exists the strong belief that a continuous Galerkin scheme is notoriously unstable and additional stabilization terms have to be added to guarantee stability. In the first part of the series [6], the application of simultaneous approximation terms for linear problems is investigated where the boundary conditions are imposed weakly. By applying this technique, the authors demonstrate that a pure continuous Galerkin scheme is indeed linearly stable if the boundary conditions are imposed in the correct way. In this work, we extend this investigation to the nonlinear case and focus on entropy conservation. By switching to entropy variables, we provide an estimation of the boundary operators also for nonlinear problems, that guarantee conservation. In numerical simulations, we verify our theoretical analysis.
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- 2021
5. A multi-domain summation-by-parts formulation for complex geometries
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Tomas Lundquist, Fredrik Laurén, and Jan Nordström
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Computational Mathematics ,Numerical Analysis ,Matematik ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Modeling and Simulation ,Summation-by-parts ,Multi-block operators ,Partial derivative approximations ,Nonlinear stability ,Mathematics ,Computer Science Applications - Abstract
We combine existing summation-by-parts discretization methods to obtain a simplified numerical framework for partial differential equations posed on complex multi-block/element domains. The interfaces (conforming or non-conforming) between blocks are treated with inner-product-preserving interpolation operators, and the result is a high-order multi-block operator on summation-by-parts form that encapsulates both the metric terms as well as the interface treatments. This enables for a compact description of the numerical scheme that mimics the essential features of its continuous counterpart. Furthermore, the stability analysis on a multi-block domain is simplified for both for linear and nonlinear equations, since no problem-specific interface conditions need to be derived and implemented. We exemplify the combined operator technique by considering a nonlinearly stable discrete formulation of the incompressible Navier-Stokes equations and perform calculations on an underlying multi-block domain. Funding: Vetenskapsradet, Sweden [2020-03642, 2018-05084, 2021-05484]; Swedish e-Science Research Center (SeRC)
- Published
- 2022
6. A skew-symmetric energy and entropy stable formulation of the compressible Euler equations
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Jan Nordström
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Numerical Analysis ,Matematik ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,65M12 ,Skew-symmetric form ,Numerical Analysis (math.NA) ,Compressible Euler equations ,Entropy stability ,Computer Science Applications ,Summation-by-parts ,Computational Mathematics ,Mathematics - Analysis of PDEs ,Modeling and Simulation ,FOS: Mathematics ,Nonlinear hyperbolic problems ,Mathematics - Numerical Analysis ,Energy stability ,Mathematics ,Analysis of PDEs (math.AP) - Abstract
We show that a specific skew-symmetric form of nonlinear hyperbolic problems leads to energy and entropy bounds. Next, we exemplify by considering the compressible Euler equations in primitive variables, transform them to skew-symmetric form and show how to obtain energy and entropy estimates. Finally we show that the skew-symmetric formulation lead to energy and entropy stable discrete approximations if the scheme is formulated on summation-by-parts form. Funding agencies: Vetenskapsradet, Sweden [2018-05084 VR]; Swedish e-Science Research Center (SeRC)
- Published
- 2022
7. On the theoretical foundation of overset grid methods for hyperbolic problems : Well-posedness and conservation
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Jan Nordström, Gregor J. Gassner, and David A. Kopriva
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Penalty methods ,Physics and Astronomy (miscellaneous) ,Scalar (mathematics) ,Conservation ,Space (mathematics) ,System of linear equations ,Domain (mathematical analysis) ,FOS: Mathematics ,Applied mathematics ,Mathematics - Numerical Analysis ,Single domain ,Mathematics ,Coupling ,Numerical Analysis ,Matematik ,Applied Mathematics ,Numerical Analysis (math.NA) ,Computer Science Applications ,Overset grids ,Computational Mathematics ,Well-posedness ,Modeling and Simulation ,Bounded function ,Chimera method ,Stability ,Energy (signal processing) - Abstract
We use the energy method to study the well-posedness of initial-boundary value problems approximated by overset mesh methods in one and two space dimensions for linear constant-coefficient hyperbolic systems. We show that in one space dimension, for both scalar equations and systems of equations, the problem where one domain partially oversets another is well-posed when characteristic coupling conditions are used. If a system cannot be diagonalized, as is usually the case in multiple space dimensions, then the energy method does not give proper bounds in terms of initial and boundary data. For those problems, we propose a novel penalty approach. We show, by using a global energy that accounts for the energy in the overlap region of the domains, that under well-defined conditions on the coupling matrices the penalized overset domain problems are energy bounded, conservative, well-posed and have solutions equivalent to the original single domain problem. Funding: Simons Foundation [426393]; Vetenskapsradet, SwedenSwedish Research Council [2018-05084 VR]; Swedish e-Science Research Center (SeRC); Klaus-Tschira Stiftung; European Research CouncilEuropean Research Council (ERC)European Commission [71448]
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- 2022
8. A linear and nonlinear analysis of the shallow water equations and its impact on boundary conditions
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Andrew Winters and Jan Nordström
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Computational Mathematics ,Numerical Analysis ,Shallow water equations ,Matematik ,Boundary conditions ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Modeling and Simulation ,Energy stability ,Nonlinear hyperbolic equations ,Entropy stability ,Mathematics ,Computer Science Applications - Abstract
We derive boundary conditions and estimates based on the energy and entropy analysis of systems of the nonlinear shallow water equations in two spatial dimensions. It is shown that the energy method provides more details, but is fully consistent with the entropy analysis. The details brought forward by the nonlinear energy analysis allow us to pinpoint where the difference between the linear and nonlinear analysis originate. We find that the result from the linear analysis does not necessarily hold in the nonlinear case. The nonlinear analysis leads in general to a different minimal number of boundary conditions compared with the linear analysis. In particular, and contrary to the linear case, the magnitude of the flow does not influence the number of required boundary conditions. Funding: Vetenskapsradet, Sweden [2018-05084 VR, 2020-03642 VR]
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- 2022
9. A stable and conservative nonlinear interface coupling for the incompressible Euler equations
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Jan Nordström and Fredrik Laurén
- Subjects
Summation-by-parts ,Matematik ,Applied Mathematics ,Incompressible Euler equations ,Nonlinear interface conditions ,Conservation ,Stability ,Mathematics - Abstract
Energy stable and conservative nonlinear weakly imposed interface conditions for the incompressible Euler equations are derived in the continuous setting. By discretely mimicking the continuous analysis using summation-by-parts operators, we prove that the numerical scheme is stable and conservative. The theoretical findings are verified by numerical experiments. Funding: Vetenskapradet [2018-05084, 2021-05484]
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- 2022
10. Convergence of energy stable finite-difference schemes with interfaces
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Magnus Svärd and Jan Nordström
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Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Beräkningsmatematik ,Applied Mathematics ,Finite difference ,010103 numerical & computational mathematics ,Finite difference method ,Grid ,01 natural sciences ,Stability (probability) ,Computer Science Applications ,010101 applied mathematics ,Computational Mathematics ,Convergence properties ,Convergence results ,Finite difference scheme ,Grid blocks ,High order finite difference schemes ,Multiple dimensions ,Single domains ,Rate of convergence ,Modeling and Simulation ,Multiple time dimensions ,Convergence (routing) ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Energy (signal processing) ,Mathematics - Abstract
We extend the convergence results in Svärd and Nordström (2019) [7] for single-domain energy-stable high-order finite difference schemes, to include domains split into several grid blocks. The analysis also demonstrates that reflective boundary conditions enjoy the same convergence properties. Finally, we briefly indicate that these results (and the previous ones in [7]) also hold in multiple dimensions. publishedVersion
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- 2021
11. Stable Dynamical Adaptive Mesh Refinement
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Jan Nordström, Arnaud G. Malan, and Tomas Lundquist
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Beräkningsmatematik ,010103 numerical & computational mathematics ,01 natural sciences ,Theoretical Computer Science ,Polygon mesh ,0101 mathematics ,Accuracy ,Mathematics ,Numerical Analysis ,Transmission problem ,Adaptive mesh refinement ,Applied Mathematics ,General Engineering ,Finite difference ,Interpolation ,010101 applied mathematics ,Computational Mathematics ,Computational Theory and Mathematics ,Transmission (telecommunications) ,Product (mathematics) ,Semi-boundedness ,Algorithm ,Stability ,Software ,Inner product preserving - Abstract
We consider accurate and stable interpolation procedures for numerical simulations utilizingtime dependent adaptive meshes. The interpolation of numerical solution valuesbetween meshes is considered as a transmission problem with respect to the underlying semidiscretizedequations, and a theoretical framework using inner product preserving operatorsis developed, which allows for both explicit and implicit implementations. The theory issupplemented with numerical experiments demonstrating practical benefits of the new stableframework. For this purpose, new interpolation operators have been designed to be used withmulti-block finite difference schemes involving non-collocated, moving interfaces. Funding:National Research Foundation of South AfricaNational Research Foundation - South Africa [89916]; Vetenskapsradet, SwedenSwedish Research Council [2018-05084_VR]
- Published
- 2021
12. Stability of Discontinuous Galerkin Spectral Element Schemes for Wave Propagation when the Coefficient Matrices have Jumps
- Author
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Jan Nordström, Gregor J. Gassner, and David A. Kopriva
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Beräkningsmatematik ,Discontinuous Galerkin spectral element ,010103 numerical & computational mathematics ,01 natural sciences ,Article ,Theoretical Computer Science ,Discontinuous Galerkin method ,FOS: Mathematics ,Boundary value problem ,Mathematics - Numerical Analysis ,0101 mathematics ,Mathematics ,Numerical Analysis ,Partial differential equation ,Applied Mathematics ,Mathematical analysis ,General Engineering ,Linear advection ,Numerical Analysis (math.NA) ,Stability ,Discontinuous coefficients ,010101 applied mathematics ,Computational Mathematics ,Computational Theory and Mathematics ,Bounded function ,Norm (mathematics) ,Dissipative system ,Hyperbolic partial differential equation ,Software ,Energy (signal processing) - Abstract
We use the behavior of the $$L_{2}$$ L 2 norm of the solutions of linear hyperbolic equations with discontinuous coefficient matrices as a surrogate to infer stability of discontinuous Galerkin spectral element methods (DGSEM). Although the $$L_{2}$$ L 2 norm is not bounded in terms of the initial data for homogeneous and dissipative boundary conditions for such systems, the $$L_{2}$$ L 2 norm is easier to work with than a norm that discounts growth due to the discontinuities. We show that the DGSEM with an upwind numerical flux that satisfies the Rankine–Hugoniot (or conservation) condition has the same energy bound as the partial differential equation does in the $$L_{2}$$ L 2 norm, plus an added dissipation that depends on how much the approximate solution fails to satisfy the Rankine–Hugoniot jump.
- Published
- 2020
13. Multigrid Schemes for High Order Discretizations of Hyperbolic Problems
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Andrea Alessandro Ruggiu and Jan Nordström
- Subjects
MathematicsofComputing_NUMERICALANALYSIS ,010103 numerical & computational mathematics ,Residual ,01 natural sciences ,Mathematics::Numerical Analysis ,Theoretical Computer Science ,High order finite difference methods ,Summation-by-parts ,Multigrid ,Hyperbolic problems ,Convergence acceleration ,Multigrid method ,Total variation diminishing ,Applied mathematics ,High order ,0101 mathematics ,Spurious relationship ,Mathematics ,Matematik ,Numerical Analysis ,Conservation law ,Applied Mathematics ,General Engineering ,Prolongation ,First order ,Computer Science::Numerical Analysis ,010101 applied mathematics ,Computational Mathematics ,Computational Theory and Mathematics ,Rewriting ,Software ,Interpolation - Abstract
Total variation diminishing multigrid methods have been developed for first order accurate discretizations of hyperbolic conservation laws. This technique is based on a so-called upwind biased residual interpolation and allows for algorithms devoid of spurious numerical oscillations in the transient phase. In this paper, we justify the introduction of such prolongation and restriction operators by rewriting the algorithm in a matrix-vector notation. This perspective sheds new light on multigrid procedures for hyperbolic problems and provides a direct extension for high order accurate difference approximations. The new multigrid procedure is presented, advantages and disadvantages are discussed and numerical experiments are performed. Funding agencies: Linkoping University; VINNOVA, the Swedish Governmental Agency for Innovation SystemsVinnova [2013-01209]
- Published
- 2020
14. The spatial operator in the incompressible Navier–Stokes, Oseen and Stokes equations
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Fredrik Laurén and Jan Nordström
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Incompressible Navier-Stokes equations ,Beräkningsmatematik ,Computational Mechanics ,Mathematics::Analysis of PDEs ,General Physics and Astronomy ,010103 numerical & computational mathematics ,01 natural sciences ,Physics::Fluid Dynamics ,Oseen equations ,Eigenvalue problem ,Boundary value problem ,0101 mathematics ,Mathematics ,Semi-bounded operators ,Matematik ,Mechanical Engineering ,Operator (physics) ,Null (mathematics) ,Mathematical analysis ,Spectrum (functional analysis) ,Computational mathematics ,Stokes equations ,Computer Science Applications ,010101 applied mathematics ,Nonlinear system ,Computational Mathematics ,Mechanics of Materials ,Compressibility ,Gravitational singularity ,Singularities - Abstract
We investigate the spatial operator in the incompressible Navier–Stokes, Oseen and Stokes equations and show how to avoid singularities associated with null spaces by choosing specific boundary conditions. The theoretical results are derived for a general form of energy stable boundary conditions, and applied to a few commonly used ones. The analysis is done on a system that simultaneously covers the nonlinear incompressible Navier–Stokes, the Oseen and the Stokes equations. When the spectrum of the spatial operator is investigated, we restrict the analysis to the Oseen and Stokes equations. The continuous analysis carries over to the discrete setting by using the summation-by-parts framework.
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- 2020
15. On conservation and dual consistency for summation-by-parts based approximations of parabolic problems
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Fatemeh Ghasemi and Jan Nordström
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Dual consistency ,Coupling ,Numerical Analysis ,Matematik ,Physics and Astronomy (miscellaneous) ,Summation by parts ,Discretization ,Approximations of π ,Applied Mathematics ,Computational mathematics ,010103 numerical & computational mathematics ,Conservation ,Weak interface conditions ,01 natural sciences ,Computer Science Applications ,010101 applied mathematics ,Summation-by-parts ,Computational Mathematics ,Modeling and Simulation ,Dual consistent ,Applied mathematics ,Parabolic problems ,0101 mathematics ,Mathematics - Abstract
We consider the coupling of parabolic problems discretized using difference operators on summation-by-parts (SBP) form with interface conditions imposed weakly. In [1, 2], it was shown that conserv ...
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- 2020
16. The relation between primal and dual boundary conditions for hyperbolic systems of equations
- Author
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Fatemeh Ghasemi and Jan Nordström
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Physics and Astronomy (miscellaneous) ,Relation (database) ,Discretization ,Computation ,Dual problem ,010103 numerical & computational mathematics ,01 natural sciences ,Hyperbolic systems ,Simple (abstract algebra) ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Scaling ,Mathematics ,Numerical Analysis ,Matematik ,Boundary conditions ,Primal problem ,Applied Mathematics ,Computational mathematics ,Dual consistency ,Computer Science Applications ,Dual (category theory) ,010101 applied mathematics ,Computational Mathematics ,Well-posedness ,Modeling and Simulation - Abstract
In this paper we study boundary conditions for linear hyperbolic systems of equations and the corresponding dual problem. In particular, we show that the primal and dual boundary conditions are related by a simple scaling relation. It is also shown that the weak dual problem can be derived directly from the weak primal problem. Based on the continuous analysis, we discretize and perform computations with a high-order finite difference scheme on summation- by-parts form with weak boundary conditions. It is shown that the results obtained in the continuous analysis lead directly to stability results for the primal and dual discrete problems. Numerical experiments corroborate the theoretical results.
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- 2020
17. Eigenvalue analysis for summation-by-parts finite difference time discretizations
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Andrea Alessandro Ruggiu and Jan Nordström
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Numerical Analysis ,Matematik ,Summation by parts ,Sixth order ,Applied Mathematics ,Diagonal ,Finite difference method ,Finite difference ,010103 numerical & computational mathematics ,01 natural sciences ,Computational Mathematics ,Eigenvalue analysis ,Norm (mathematics) ,Initial value problem ,Applied mathematics ,0101 mathematics ,Mathematics ,time integration ,initial value problem ,summation-by-parts operators ,finite difference methods ,eigenvalue problem - Abstract
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 Runge--Kutta methods based on second, fourth, and sixth order summation-by-parts finite difference time discretizations automatically satisfy previously unreported stability properties. The procedure outlined in this article can be extended to even higher order summation-by-parts approximations with repeating stencil. Funding agencies: VINNOVA, the Swedish Governmental Agency for Innovation SystemsVinnova [2013-01209]
- Published
- 2020
18. The Number of Boundary Conditions for Initial Boundary Value Problems
- Author
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Jan Nordström and Thomas Hagstrom
- Subjects
Numerical Analysis ,Matematik ,Summation by parts ,Laplace transform ,Applied Mathematics ,Mathematical analysis ,initial boundary value problems ,Computational mathematics ,energy method ,010103 numerical & computational mathematics ,01 natural sciences ,Computational Mathematics ,Laplace transform method ,Normal mode ,incompletely parabolic ,normal mode analysis ,summation-by-parts ,boundary conditions ,Energy method ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
Both the energy method and the Laplace transform method are frequently used for determining the number of boundary conditions required for a well posed initial boundary value problem. We show that these two distinctly different methods yield the same results. The continuous energy method can be mimicked exactly in the corresponding semidiscrete problems discretized using the summation-by-parts technique. Hence the analysis of well posedness and stability can bypass the more unwieldy Laplace transform method. Funding agencies: The first author was supported by Vetenskapsrådet, Sweden grant 2018-05084 VR. The second author was supported by National Science Foundation grant DMS-2012296.
- Published
- 2020
19. Stable and Accurate Filtering Procedures
- Author
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Jan Nordström and Tomas Lundquist
- Subjects
Beräkningsmatematik ,High frequency oscillations ,010103 numerical & computational mathematics ,01 natural sciences ,Stability (probability) ,Stencil ,Theoretical Computer Science ,Control theory ,Boundary value problem ,0101 mathematics ,Numerical filters ,Accuracy ,High wave number ,Mathematics ,Transmission problem ,Matematik ,Numerical Analysis ,Semi-bounded ,Applied Mathematics ,General Engineering ,Computational mathematics ,Filter (signal processing) ,Dissipation ,Summation-by-parts ,010101 applied mathematics ,Computational Mathematics ,Computational Theory and Mathematics ,Stability ,Software ,Energy (signal processing) - Abstract
High frequency errors are always present in numerical simulations since no difference stencil is accurate in the vicinity of the $$\pi $$π-mode. To remove the defective high wave number information from the solution, artificial dissipation operators or filter operators may be applied. Since stability is our main concern, we are interested in schemes on summation-by-parts (SBP) form with weak imposition of boundary conditions. Artificial dissipation operators preserving the accuracy and energy stability of SBP schemes are available. However, for filtering procedures it was recently shown that stability problems may occur, even for originally energy stable (in the absence of filtering) SBP based schemes. More precisely, it was shown that even the sharpest possible energy bound becomes very weak as the number of filtrations grow. This suggest that successful filtering include a delicate balance between the need to remove high frequency oscillations (filter often) and the need to avoid possible growth (filter seldom). We will discuss this problem and propose a remedy.
- Published
- 2020
20. Robust boundary conditions for stochastic incompletely parabolic systems of equations
- Author
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Jan Nordström and Markus Wahlsten
- Subjects
Matematik ,Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Applied Mathematics ,Mathematical analysis ,Boundary (topology) ,Computational mathematics ,010103 numerical & computational mathematics ,Mixed boundary condition ,Uncertainty quantification ,Incompletely parabolic system ,Initial boundary value problems ,Stochastic data ,Variance reduction ,Robust design ,Space (mathematics) ,System of linear equations ,01 natural sciences ,Computer Science Applications ,010101 applied mathematics ,Computational Mathematics ,Boundary conditions in CFD ,Modeling and Simulation ,Boundary value problem ,0101 mathematics ,Mathematics ,Computer Science::Databases - Abstract
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. Funding agencies: European Commission [ACP3-GA-2013-605036]
- Published
- 2018
21. Practical inlet boundary conditions for internal flow calculations
- Author
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Jan Nordström and Fredrik Laurén
- Subjects
General Computer Science ,Beräkningsmatematik ,Boundary (topology) ,010103 numerical & computational mathematics ,Inflow ,01 natural sciences ,inlet boundary conditions ,symbols.namesake ,well-posedness ,steady state ,Boundary value problem ,0101 mathematics ,Total pressure ,Mathematics ,Internal flow ,Mathematical analysis ,General Engineering ,Euler equations ,010101 applied mathematics ,Computational Mathematics ,Nonlinear system ,Rate of convergence ,eigenmode analysis ,symbols - Abstract
To impose boundary conditions, data at the boundaries must be known, and consequently measurements of the imposed quantities must be available. In this paper, we consider the two most commonly used inflow boundary conditions with available data for internal flow calculations: the specification of the total temperature and total pressure. We use the energy method to prove that the specification of the total temperature and the total pressure together with the tangential velocity at an inflow boundary lead to well-posedness for the linearized compressible Euler equations. Next, these equations are discretized in space using high-order finite-difference operators on summation-by-parts form, and the boundary conditions are weakly imposed. The resulting numerical scheme is proven to be stable and the implementation of the corresponding nonlinear scheme is verified with the method of manufactured solutions. We also derive the spectrum for the continuous and discrete problems and show how to predict the convergence rate to steady state. Finally, nonlinear steady-state computations are performed, and they confirm the predicted convergence rates.
- Published
- 2018
22. Analysis of the SBP-SAT Stabilization for Finite Element Methods Part I: Linear Problems
- Author
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Jan Nordström, Svetlana Tokareva, Rémi Abgrall, Philipp Öffner, University of Zurich, and Öffner, Philipp
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Beräkningsmatematik ,340 Law ,Boundary (topology) ,610 Medicine & health ,010103 numerical & computational mathematics ,01 natural sciences ,Article ,Theoretical Computer Science ,Initial-boundary value problem ,symbols.namesake ,510 Mathematics ,2604 Applied Mathematics ,Discontinuous Galerkin method ,FOS: Mathematics ,Simultaneous approximation terms ,Applied mathematics ,Mathematics - Numerical Analysis ,Boundary value problem ,0101 mathematics ,2614 Theoretical Computer Science ,Galerkin method ,2612 Numerical Analysis ,Mathematics ,Numerical Analysis ,Applied Mathematics ,General Engineering ,Finite difference ,Numerical Analysis (math.NA) ,Continuous Galerkin ,Finite element method ,1712 Software ,010101 applied mathematics ,10123 Institute of Mathematics ,Computational Mathematics ,Discontinuity (linguistics) ,Computational Theory and Mathematics ,65M12, 65M60, 65M70 ,Hyperbolic conservation laws ,2200 General Engineering ,symbols ,Gaussian quadrature ,2605 Computational Mathematics ,Stability ,Software ,1703 Computational Theory and Mathematics - Abstract
In the hyperbolic community, discontinuous Galerkin approaches are mainly applied when finite element methods are considered. As the name suggested, the DG framework allows a discontinuity at the element interfaces, which seems for many researchers a favorable property in case of hyperbolic balance laws. On the contrary, continuous Galerkin method obtained from a straightforward discretisation of the weak form of the PDEs appear to be unsuitable for hyperbolic problems. To remedy this issue, stabilization terms are usually added and various formulations can be found in the literature. There exists still the perception that continuous Galerkin methods are not suited to hyperbolic problems, and the reason of this is the continuity of the approximation. However, this perception is not true and the stabilization terms can be removed, in general, provided the boundary conditions are suitable. In this paper, we deal with this problem, and present a different approach. We use the boundary conditions to stabilize the scheme following a procedure that are frequently used in the finite difference community. Here, the main idea is to impose the boundary conditions weakly and specific boundary operators are constructed such that they guarantee stability. This approach has already been used in the DG framework, but here we apply it with a continuous Galerkin scheme. No internal dissipation is needed even if unstructured grids are used. Further, we point out that we do not need exact integration, it suffices if the quadrature rule and the norm in the differential operator are the same, such that the summation-by-parts (SBP) property is fulfilled meaning that a discrete Gauss Th. is valid. This contradicts the perception in the hyperbolic community that stability issues for pure Galerkin scheme exist. In numerical simulations, we verify our theoretical analysis., 28 pages, 10 figures
- Published
- 2019
23. Well-posed and stable transmission problems
- Author
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Viktor Linders and Jan Nordström
- Subjects
Well-posed problem ,Matematik ,Numerical Analysis ,Class (set theory) ,Multi grid ,Physics and Astronomy (miscellaneous) ,Adaptive mesh refinement ,Applied Mathematics ,Stability (learning theory) ,Computational mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Computer Science Applications ,010101 applied mathematics ,Transmission problems ,Well-posedness ,Stability ,Numerical filter ,Multi-grid ,Computational Mathematics ,Transmission (telecommunications) ,Modeling and Simulation ,Calculus ,Applied mathematics ,0101 mathematics ,Mathematics ,Well posedness - Abstract
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. Funding agencies: Swedish Meteorological and Hydrological Institute (SMHI)
- Published
- 2018
24. A new multigrid formulation for high order finite difference methods on summation-by-parts form
- Author
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Jan Nordström, Andrea Alessandro Ruggiu, and Per Weinerfelt
- Subjects
Matematik ,Numerical Analysis ,Convergence acceleration ,Physics and Astronomy (miscellaneous) ,Summation by parts ,Applied Mathematics ,Mathematical analysis ,Finite difference method ,Computational mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,High order finite difference methodsSummation-by-partsMultigridRestriction and prolongation operatorsConvergence acceleration ,Computer Science Applications ,010101 applied mathematics ,Computational Mathematics ,Multigrid method ,Modeling and Simulation ,Applied mathematics ,0101 mathematics ,High order ,Mathematics ,Interpolation - Abstract
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. Funding agencies: VINNOVA, the Swedish Governmental Agency for Innovation Systems [2013-01209]
- Published
- 2018
25. On Long Time Error Bounds for the Wave Equation on Second Order Form
- Author
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Hannes Frenander and Jan Nordström
- Subjects
010103 numerical & computational mathematics ,Second order form ,Long times ,01 natural sciences ,Theoretical Computer Science ,Simultaneous approximation terms ,Boundary value problem ,0101 mathematics ,Mathematics ,Finite differences ,Matematik ,Numerical Analysis ,Spacetime ,Applied Mathematics ,Mathematical analysis ,General Engineering ,Computational mathematics ,Wave equation ,Summation-by-parts ,010101 applied mathematics ,Error bounds ,Computational Mathematics ,Order form ,Computational Theory and Mathematics ,Time error ,Software - Abstract
Temporal error bounds for the wave equation expressed on second order form are investigated. We show that, with appropriate choices of boundary conditions, the time and space derivatives of the error are bounded even for long times. No long time bound on the error itself is obtained, although numerical experiments indicate that a bound exists.
- Published
- 2018
26. Spurious solutions for the advection-diffusion equation using wide stencils for approximating the second derivative
- Author
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Jan Nordström and Hannes Frenander
- Subjects
Matematik ,Numerical Analysis ,Summation by parts ,Truncation error (numerical integration) ,Applied Mathematics ,Mathematical analysis ,oscillating solutions ,010103 numerical & computational mathematics ,01 natural sciences ,Stencil ,Stability (probability) ,010101 applied mathematics ,Computational Mathematics ,spurious solutions ,Rate of convergence ,summation-by-parts ,0101 mathematics ,Convection–diffusion equation ,Spurious relationship ,Mathematics ,Analysis ,Second derivative - Abstract
A one-dimensional steady-state advection-diffusion problem using summation-by-parts operators is investigated. For approximating the second derivative, a wide stencil is used, which simplifies implementation and stability proofs. However, it also introduces spurious, oscillating, modes for all mesh sizes. We prove that the size of the spurious modes is equal to the size of the truncation error for a stable approximation and hence disappears with the convergence rate. The theoretical results are verified with numerical experiments. Funding agencies:This project was funded by the Swedishe-science Research Center (SeRC). Thefunding source had no involvement in thestudy design, collection and analysis ofdata, or in writing and submitting thisarticle
- Published
- 2017
27. Coupling Requirements for Multiphysics Problems Posed on Two Domains
- Author
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Fatemeh Ghasemi and Jan Nordström
- Subjects
well posed problems ,Mathematical optimization ,Beräkningsmatematik ,Multiphysics ,Stability (learning theory) ,high order finite diffrences ,010103 numerical & computational mathematics ,01 natural sciences ,dual consistency ,stiffness ,summation-by-parts ,Applied mathematics ,0101 mathematics ,Mathematics ,weak interface conditions ,Numerical Analysis ,Summation by parts ,Applied Mathematics ,Computational mathematics ,stability ,Superconvergence ,First order ,Hyperbolic systems ,010101 applied mathematics ,superconvergence ,Computational Mathematics ,Coupling (physics) - Abstract
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.
- Published
- 2017
28. Eigenvalue analysis and convergence acceleration techniques for summation-by-parts approximations
- Author
-
Jan Nordström and Andrea Alessandro Ruggiu
- Subjects
Partial differential equation ,Convergence acceleration ,Summation by parts ,Eigenvalue analysis ,Physical phenomena ,Bounded function ,Mathematics::Metric Geometry ,Applied mathematics ,Computational mathematics ,Computer Science::Databases ,Mathematics - Abstract
Many physical phenomena can be described mathematically by means of partial differential equations. These mathematical formulations are said to be well-posed if a unique solution, bounded by the gi ...
- Published
- 2019
29. On Stochastic Investigation of Flow Problems Using the Viscous Burgers’ Equation as an Example
- Author
-
Jan Nordström and Markus Wahlsten
- Subjects
Beräkningsmatematik ,MathematicsofComputing_NUMERICALANALYSIS ,01 natural sciences ,Projection (linear algebra) ,010305 fluids & plasmas ,Theoretical Computer Science ,0103 physical sciences ,Applied mathematics ,0101 mathematics ,Uncertainty quantification ,Mathematics ,Computer Science::Cryptography and Security ,Numerical Analysis ,Polynomial chaos ,Stochastic process ,Applied Mathematics ,General Engineering ,Computational mathematics ,Numerical integration ,Burgers' equation ,Stochastic data ,Stochastic Galerkin ,Intrusive methods ,Non-intrusive methods ,Burgers’ equation ,010101 applied mathematics ,Computational Mathematics ,Computational Theory and Mathematics ,Flow (mathematics) ,Software - Abstract
We consider a stochastic analysis of non-linear viscous fluid flow problems with smooth and sharp gradients in stochastic space. As a representative example we consider the viscous Burgers’ equation and compare two typical intrusive and non-intrusive uncertainty quantification methods. The specific intrusive approach uses a combination of polynomial chaos and stochastic Galerkin projection. The specific non-intrusive method uses numerical integration by combining quadrature rules and the probability density functions of the prescribed uncertainties. The two methods are compared in terms of error in the estimated variance, computational efficiency and accuracy. This comparison, although not general, provide insight into uncertainty quantification of problems with a combination of sharp and smooth variations in stochastic space. It suggests that combining intrusive and non-intrusive methods could be advantageous.
- Published
- 2019
30. Energy Stable Boundary Conditions for the Nonlinear Incompressible Navier-Stokes Equations
- Author
-
Jan Nordström and Cristina La Cognata
- Subjects
Matematik ,Algebra and Number Theory ,high-order accuracy ,energy estimate ,Applied Mathematics ,incompressible ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Mixed boundary condition ,Different types of boundary conditions in fluid dynamics ,stability ,divergence free ,Robin boundary condition ,Physics::Fluid Dynamics ,Computational Mathematics ,Boundary conditions in CFD ,summation-by-parts ,boundary conditions ,Free boundary problem ,Neumann boundary condition ,Boundary value problem ,Navier-Stokes equations ,Navier–Stokes equations ,Mathematics - Abstract
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.
- Published
- 2019
31. Hybrid Computational-Fluid-Dynamics Platform to Investigate Aircraft Trailing Vortices
- Author
-
Arnaud G. Malan, Donovan M. Changfoot, and Jan Nordström
- Subjects
Physics ,020301 aerospace & aeronautics ,Lift coefficient ,Matematik ,Finite volume method ,business.industry ,Finite difference ,Aerospace Engineering ,Upwind scheme ,02 engineering and technology ,Mechanics ,Computational fluid dynamics ,01 natural sciences ,010305 fluids & plasmas ,Vortex ,0203 mechanical engineering ,0103 physical sciences ,No-slip condition ,Navier–Stokes equations ,business ,Mathematics - Abstract
This paper outlines the development of a parallel three-dimensional hybrid finite volume finite difference capability. The specific application area under consideration is modeling the trailing vortices shed from the wings of aircraft under transonic flight conditions. For this purpose, the Elemental finite volume code is employed in the vicinity of the aircraft, whereas the ESSENSE finite difference software is employed to accurately resolve the trailing vortices. The former method is spatially formally second-order, and the latter is set to sixth-order accuracy. The coupling of the two methods is achieved in a stable manner through the use of summation-by-parts operators and weak imposition of boundary conditions using simultaneous approximation terms. The developed hybrid solver is successfully validated against an analytical test case. This is followed by demonstrating the ability to model the flowfield, including trailing vortex structures, around the NASA Common Research Model under transonic flow conditions. The interface treatment is shown to describe the intersecting vortices in a smooth manner. In addition, insights gained in resolving the vortices include violation of underlying assumptions of analytical vortex modeling methods. Funding agencies: National Aerospace Centre of the University of Witwatersrand, Johannesburg; South African Research (SARChI) Chair in Industrial CFD - Department of Science and Technology; South African Research (SARChI) Chair in Industrial CFD - National Research Foundation
- Published
- 2019
32. Galerkin Projection and Numerical Integration for a Stochastic Investigation of the Viscous Burgers’ Equation: An Initial Attempt
- Author
-
Jan Nordström and Markus Wahlsten
- Subjects
Polynomial chaos ,Stochastic process ,MathematicsofComputing_NUMERICALANALYSIS ,Applied mathematics ,Computational mathematics ,Uncertainty quantification ,Galerkin method ,Computer Science::Cryptography and Security ,Burgers' equation ,Mathematics ,Quadrature (mathematics) ,Numerical integration - Abstract
We consider a stochastic analysis of the non-linear viscous Burgers’ equation and focus on the comparison between intrusive and non-intrusive uncertainty 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.
- Published
- 2019
33. Accuracy of Stable, High-order Finite Difference Methods for Hyperbolic Systems with Non-smooth Wave Speeds
- Author
-
Ossian O'Reilly, Jan Nordström, and Brittany A. Erickson
- Subjects
Numerical Analysis ,Matematik ,Laplace transform ,Advection ,Applied Mathematics ,Scalar (mathematics) ,Mathematical analysis ,General Engineering ,Finite difference method ,Order of accuracy ,Classification of discontinuities ,Instability ,Theoretical Computer Science ,Computational Mathematics ,Computational Theory and Mathematics ,Boundary value problem ,Software ,Mathematics - Abstract
We derive analytic solutions to the scalar and vector advection equation with variable coefficients in one spatial dimension using Laplace transform methods. These solutions are used to investigate how accuracy and stability are influenced by the presence of discontinuous wave speeds when applying high-order-accurate, skew-symmetric finite difference methods designed for smooth wave speeds. The methods satisfy a summation-by-parts rule with weak enforcement of boundary conditions and formal order of accuracy equal to 2, 3, 4 and 5. We study accuracy, stability and convergence rates for linear wave speeds that are (a) constant, (b) non-constant but smooth, (c) continuous with a discontinuous derivative, and (d) constant with a jump discontinuity. Cases (a) and (b) correspond to smooth wave speeds and yield stable schemes and theoretical convergence rates. Non-smooth wave speeds [cases (c) and (d)], however, reveal reductions in theoretical convergence rates and in the latter case, the presence of an instability.
- Published
- 2019
34. An energy stable coupling procedure for the compressible and incompressible Navier-Stokes equations
- Author
-
Jan Nordström and Fatemeh Ghasemi
- Subjects
Well-posed problem ,Physics and Astronomy (miscellaneous) ,Discretization ,Interface (Java) ,010103 numerical & computational mathematics ,01 natural sciences ,Stability (probability) ,Compressible flow ,Energy estimate ,Incompressible fluid ,0101 mathematics ,Navier–Stokes equations ,Physics ,Coupling ,Numerical Analysis ,Matematik ,Applied Mathematics ,Mathematical analysis ,Computational mathematics ,Computer Science Applications ,010101 applied mathematics ,Computational Mathematics ,Modeling and Simulation ,Interface conditions ,Navier-Stokes equations ,Compressible fluid ,Stability ,Mathematics - Abstract
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.
- Published
- 2019
35. Dual Time-Stepping Using Second Derivatives
- Author
-
Andrea Alessandro Ruggiu and Jan Nordström
- Subjects
Numerical Analysis ,Matematik ,Current (mathematics) ,Applied Mathematics ,General Engineering ,Stiffness ,010103 numerical & computational mathematics ,01 natural sciences ,Theoretical Computer Science ,Dual (category theory) ,010101 applied mathematics ,Computational Mathematics ,Operator (computer programming) ,Computational Theory and Mathematics ,Square root ,Time stepping ,Convergence (routing) ,medicine ,Applied mathematics ,0101 mathematics ,medicine.symptom ,Software ,Mathematics ,Second derivative - Abstract
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. Funding agencies: Linkoping University; Swedish Governmental Agency for Innovation SystemsVinnova [2013-01209]; VINNOVAVinnova
- Published
- 2019
36. Hyperbolic systems of equations posed on erroneous curved domains
- Author
-
Samira Nikkar and Jan Nordström
- Subjects
Matematik ,Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Discretization ,Applied Mathematics ,Mathematical analysis ,Boundary (topology) ,Order of accuracy ,Geometry ,010103 numerical & computational mathematics ,Hyperbolic systems ,Erroneous curved domains ,Inaccurate data ,Convergence rate ,01 natural sciences ,Computer Science Applications ,Zero (linguistics) ,010101 applied mathematics ,Computational Mathematics ,Operator (computer programming) ,Rate of convergence ,Modeling and Simulation ,Imperfect ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
The effect of an inaccurate geometry description on the solution accuracy of a hyperbolic problem is discussed. The inaccurate geometry can for example come from an imperfect CAD system, a faulty mesh generator, bad measurements or simply a misconception. We show that inaccurate geometry descriptions might lead to the wrong wave speeds, a misplacement of the boundary conditions, to the wrong boundary operator and a mismatch of boundary data. The errors caused by an inaccurate geometry description may affect the solution more than the accuracy of the specific discretization techniques used. In extreme cases, the order of accuracy goes to zero. Numerical experiments corroborate the theoretical results.
- Published
- 2016
37. Summation-By-Parts in Time: The Second Derivative
- Author
-
Tomas Lundquist and Jan Nordström
- Subjects
Beräkningsmatematik ,initial boundary value problems ,010103 numerical & computational mathematics ,weak initial conditions ,01 natural sciences ,second derivative approximation ,boundary conditions ,Convergence (routing) ,Initial value problem ,Boundary value problem ,0101 mathematics ,summation-by-parts operators ,Mathematics ,Second derivative ,convergence ,Summation by parts ,time integration ,finite difference ,Applied Mathematics ,Mathematical analysis ,Finite difference ,Computational mathematics ,high order accuracy ,stability ,Wave equation ,010101 applied mathematics ,Computational Mathematics ,second order form ,wave equation ,initial value problem - Abstract
We analyze the extension of summation-by-parts operators and weak boundary conditions for solving initial boundary value problems involving second derivatives in time. A wide formulation is obtained by first rewriting the problem on first order form. This formulation leads to optimally sharp fully discrete energy estimates that are unconditionally stable and high order accurate. Furthermore, it provides a natural way to impose mixed boundary conditions of Robin type, including time and space derivatives. We apply the new formulation to the wave equation and derive optimal fully discrete energy estimates for general Robin boundary conditions, including nonreflecting ones. The scheme utilizes wide stencil operators in time, whereas the spatial operators can have both wide and compact stencils. Numerical calculations verify the stability and accuracy of the method. We also include a detailed discussion on the added complications when using compact operators in time and give an example showing that an energy estimate cannot be obtained using a standard second order accurate compact stencil. Funding agencies: Swedish Research Council [621-2012-1689]
- Published
- 2016
38. Efficient fully discrete summation-by-parts schemes for unsteady flow problems
- Author
-
Tomas Lundquist and Jan Nordström
- Subjects
Matematik ,Summation by parts ,Discretization ,Computer Networks and Communications ,Applied Mathematics ,Mathematical analysis ,Diagonal ,Summation-by-parts in time – Unsteady flow calculations – Temporal efficiency ,Finite difference ,Computational mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Stability (probability) ,010101 applied mathematics ,Computational Mathematics ,Boundary layer ,Boundary value problem ,0101 mathematics ,Mathematics ,Software - Abstract
We make an initial investigation into the temporal efficiency of a fully discrete summation-by-parts approach for unsteady flows. As a model problem for the Navier–Stokes equations we consider a two-dimensional advection–diffusion problem with a boundary layer. The problem is discretized in space using finite difference approximations on summation-by-parts form together with weak boundary conditions, leading to optimal stability estimates. For the time integration part we consider various forms of high order summation-by-parts operators and compare with an existing popular fourth order diagonally implicit Runge–Kutta method. To solve the resulting fully discrete equation system, we employ a multi-grid scheme with dual time stepping.
- Published
- 2015
39. Uniformly best wavenumber approximations by spatial central difference operators
- Author
-
Jan Nordström and Viktor Linders
- Subjects
Numerical Analysis ,Approximation theory ,Physics and Astronomy (miscellaneous) ,Continuous function ,Basis (linear algebra) ,Dispersion relation ,Wave propagation ,Wavenumber approximation ,Finite differences ,Beräkningsmatematik ,Applied Mathematics ,Mathematical analysis ,Finite difference ,Computer Science Applications ,Computational Mathematics ,Uniform norm ,Modeling and Simulation ,Wavenumber ,Subspace topology ,Mathematics - Abstract
We construct accurate central difference stencils for problems involving high frequency waves or multi-frequency solutions over long time intervals with a relatively coarse spatial mesh, and with an easily obtained bound on the dispersion error. This is done by demonstrating that the problem of constructing central difference stencils that have minimal dispersion error in the infinity norm can be recast into a problem of approximating a continuous function from a finite dimensional subspace with a basis forming a Chebyshev set. In this new formulation, characterising and numerically obtaining optimised schemes can be done using established theory.
- Published
- 2015
40. On the relation between conservation and dual consistency for summation-by-parts schemes
- Author
-
Jan Nordström and Fatemeh Ghasemi
- Subjects
Dual consistency ,Matematik ,Numerical Analysis ,Mathematical optimization ,Physics and Astronomy (miscellaneous) ,Summation by parts ,Relation (database) ,Applied Mathematics ,010103 numerical & computational mathematics ,Initial boundary value problems Summation-by-parts Conservation ,01 natural sciences ,Multi element ,Computer Science Applications ,010101 applied mathematics ,Algebra ,Computational Mathematics ,Modeling and Simulation ,Dual consistent ,Multi-block ,Multi-element ,0101 mathematics ,Mathematics - Abstract
n/a Classified in the journal as "Short note"
- Published
- 2017
41. Accurate solution-adaptive finite difference schemes for coarse and fine grids
- Author
-
Viktor Linders, Jan Nordström, and Mark H. Carpenter
- Subjects
Current (mathematics) ,Physics and Astronomy (miscellaneous) ,Truncation error (numerical integration) ,010103 numerical & computational mathematics ,01 natural sciences ,Least squares ,Convergence (routing) ,Applied mathematics ,0101 mathematics ,Accuracy ,Mathematics ,Finite differences ,Matematik ,Numerical Analysis ,Applied Mathematics ,Finite difference ,Computational mathematics ,Grid ,Computer Science Applications ,010101 applied mathematics ,Adaptivity ,Computational Mathematics ,Rate of convergence ,Dispersion relation preserving ,Modeling and Simulation ,Convergence - Abstract
We introduce solution dependent finite difference stencils whose coefficients adapt to the current numerical solution by minimizing the truncation error in the least squares sense. The resulting scheme has the resolution capacity of dispersion relation preserving difference stencils in under-resolved domains, together with the high order convergence rate of conventional central difference methods in well resolved regions. Numerical experiments reveal that the new stencils outperform their conventional counterparts on all grid resolutions from very coarse to very fine.
- Published
- 2020
42. Robust Design of Initial Boundary Value Problems
- Author
-
Markus Wahlsten and Jan Nordström
- Subjects
symbols.namesake ,Robust design ,Euler's formula ,symbols ,Boundary (topology) ,Applied mathematics ,Variance reduction ,Boundary value problem ,Variance (accounting) ,Uncertainty quantification ,Hyperbolic systems ,Mathematics - Abstract
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.
- Published
- 2018
43. Finite difference schemes with transferable interfaces for parabolic problems
- Author
-
Sofia Eriksson and Jan Nordström
- Subjects
Dual consistency ,Physics and Astronomy (miscellaneous) ,Interfaces ,010103 numerical & computational mathematics ,01 natural sciences ,Superconvergence ,Applied mathematics ,0101 mathematics ,Mathematics ,Second derivative ,Numerical Analysis ,Matematik ,Summation by parts ,Applied Mathematics ,Finite difference method ,Finite difference ,Order of accuracy ,Computational mathematics ,Computer Science Applications ,010101 applied mathematics ,Summation-by-parts ,Computational Mathematics ,Modeling and Simulation ,Finite difference methods ,High order accuracy - Abstract
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.
- Published
- 2018
44. On the order of Accuracy of Finite Difference Operators on Diagonal Norm Based Summation-by-Parts Form
- Author
-
Jan Nordström, Tomas Lundquist, and Viktor Linders
- Subjects
Diagonal ,010103 numerical & computational mathematics ,01 natural sciences ,Regular grid ,Applied mathematics ,0101 mathematics ,numerical differentiation ,summation-by-parts operators ,order of accuracy ,Computer Science::Distributed, Parallel, and Cluster Computing ,Mathematics ,Numerical Analysis ,Matematik ,Summation by parts ,Applied Mathematics ,Mathematical analysis ,Order of accuracy ,Computational mathematics ,Finite difference coefficient ,finite dierence schemes ,010101 applied mathematics ,Computational Mathematics ,Norm (mathematics) ,Numerical differentiation ,quadrature rules - Abstract
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.
- Published
- 2018
45. Summation-by-Parts Operators for Non-Simply Connected Domains
- Author
-
Jan Nordström and Samira Nikkar
- Subjects
Pure mathematics ,Matematik ,Partial differential equation ,Summation by parts ,Applied Mathematics ,non-simply connected domains ,complex geometries ,initial boundary value problems ,Stability (learning theory) ,Computational mathematics ,010103 numerical & computational mathematics ,Construct (python library) ,stability ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,well-posedness ,Simply connected space ,boundary conditions ,Boundary value problem ,0101 mathematics ,Well posedness ,Mathematics - Abstract
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.
- Published
- 2018
46. Corrigendum to 'On the relation between conservation and dual consistency for summation-by-parts schemes'[J. Comput. Phys. 344 (2017) 437–439]
- Author
-
Fatemeh Ghasemi and Jan Nordström
- Subjects
Dual consistency ,Numerical Analysis ,Matematik ,Physics and Astronomy (miscellaneous) ,Summation by parts ,Relation (database) ,Applied Mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Computer Science Applications ,010101 applied mathematics ,Computational Mathematics ,Modeling and Simulation ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
Corrigendum to “On the relation between conservation and dual consistency for summation-by-parts schemes”[J. Comput. Phys. 344 (2017) 437–439]
- Published
- 2018
47. Response to 'Convergence of Summation-by-Parts Finite Difference Methods for the Wave Equation'
- Author
-
Jan Nordström and Magnus Svärd
- Subjects
Numerical Analysis ,Matematik ,Summation by parts ,Approximations of π ,Applied Mathematics ,Mathematical analysis ,General Engineering ,Finite difference method ,Order of accuracy ,010103 numerical & computational mathematics ,Wave equation ,01 natural sciences ,Theoretical Computer Science ,010101 applied mathematics ,Computational Mathematics ,Computational Theory and Mathematics ,Convergence (routing) ,0101 mathematics ,Software ,Mathematics - Abstract
This is a short response to some statements by Wang and Kreiss in “Convergence of Summation-by-Parts Finite Difference Methods for the Wave equation” (Wang and Kreiss in J Sci Comput 71(1):219–245, 2017) that questions results in our paper “On the order of accuracy for difference approximations of initial-boundary value problems” (Svard and Nordstrom in J Comput Phys 218(1):333–352, 2006). We show that our results still stand.
- Published
- 2018
48. The effect of uncertain geometries on advection–diffusion of scalar quantities
- Author
-
Jan Nordström and Markus Wahlsten
- Subjects
Computer Networks and Communications ,Scalar (mathematics) ,Probability density function ,010103 numerical & computational mathematics ,01 natural sciences ,Variable coefficient ,Incompressible flow ,Heat transfer ,Advection–diffusion ,Parabolic problems ,Boundary value problem ,0101 mathematics ,Randomness ,Uncertainty quantification ,Mathematics ,Matematik ,Boundary conditions ,Applied Mathematics ,Mathematical analysis ,Finite difference ,Quadrature (mathematics) ,010101 applied mathematics ,Temperature field ,Computational Mathematics ,Uncertain geometry ,Random variable ,Software - Abstract
The two dimensional advection–diffusion equation in a stochastically varyinggeometry is considered. The varying domain is transformed into a fixed one andthe numerical solution is computed using a high-order finite difference formulationon summation-by-parts form with weakly imposed boundary conditions. Statistics ofthe solution are computed non-intrusively using quadrature rules given by the probabilitydensity function of the random variable. As a quality control, we prove that thecontinuous problem is strongly well-posed, that the semi-discrete problem is stronglystable and verify the accuracy of the scheme. The technique is applied to a heat transferproblem in incompressible flow. Statistical properties such as confidence intervals andvariance of the solution in terms of two functionals are computed and discussed. Weshow that there is a decreasing sensitivity to geometric uncertainty as we graduallylower the frequency and amplitude of the randomness. The results are less sensitiveto variations in the correlation length of the geometry. Funding agencies: European Union [ACP3-GA-2013-605036]
- Published
- 2018
49. On pseudo-spectral time discretizations in summation-by-parts form
- Author
-
Andrea Alessandro Ruggiu and Jan Nordström
- Subjects
Numerical Analysis ,Matematik ,Collocation ,Physics and Astronomy (miscellaneous) ,Summation by parts ,Applied Mathematics ,Order (ring theory) ,010103 numerical & computational mathematics ,01 natural sciences ,Computer Science Applications ,law.invention ,010101 applied mathematics ,Time integration ,Initial boundary value problem ,Summation-by-parts operators ,Pseudo-spectral methods ,Eigenvalue problem ,Computational Mathematics ,Invertible matrix ,law ,Modeling and Simulation ,Time derivative ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
Fully-implicit discrete formulations in summation-by-parts form for initial-boundary value problems must be invertible in order to provide well functioning procedures. We prove that, under mild assumptions, pseudo-spectral collocation methods for the time derivative lead to invertible discrete systems when energy-stable spatial discretizations are used.
- Published
- 2018
50. A Stable Domain Decomposition Technique for Advection–Diffusion Problems
- Author
-
Oskar Ålund and Jan Nordström
- Subjects
Numerical Analysis ,Matematik ,Advection ,Applied Mathematics ,General Engineering ,Domain decomposition methods ,010103 numerical & computational mathematics ,System of linear equations ,01 natural sciences ,Partial differential equations ,Theoretical Computer Science ,010101 applied mathematics ,Computational Mathematics ,Computational Theory and Mathematics ,Numerical time integration ,Finite difference methods ,Summation-by-Parts ,Applied mathematics ,Domain decomposition ,0101 mathematics ,Diffusion (business) ,Stability ,Software ,Mathematics - Abstract
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.
- Published
- 2018
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