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  • 1.
    Abbas, Qaisar
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Weak Boundary and Interface Procedures for Wave and Flow Problems2011Doctoral thesis, monograph (Other academic)
    Abstract [en]

    In this thesis, we have analyzed the accuracy and stability aspects of weak boundary and interface conditions (WBCs) for high order finite difference methods on Summations-By-Parts (SBP) form. The numerical technique has been applied to wave propagation and flow problems.

    The advantage of WBCs over strong boundary conditions is that stability of the numerical scheme can be proven. The boundary procedures in the advection-diffusion equation for a boundary layer problem is analyzed. By performing Navier-Stokes calculations, it is shown that most of the conclusions from the model problem carries over to the fully nonlinear case.

    The work was complemented to include the new idea of using WBCs on multiple grid points in a region, where the data is known, instead of at a single point. It was shown that we can achieve high accuracy, an increased rate of convergence to steady-state and non-reflecting boundary conditions by using this approach.

    Using the SBP technique and WBCs, we have worked out how to construct conservative and energy stable hybrid schemes for shocks using two different approaches. In the first method, we combine a high order finite difference scheme with a second order MUSCL scheme. In the second method, a procedure to locally change the order of accuracy of the finite difference schemes is developed. The main purpose is to obtain a higher order accurate scheme in smooth regions and a low order non-oscillatory scheme in the vicinity of shocks.

    Furthermore, we have analyzed the energy stability of the MUSCL scheme, by reformulating the scheme in the framework of SBP and artificial dissipation operators. It was found that many of the standard slope limiters in the MUSCL scheme do not lead to a negative semi-definite dissipation matrix, as required to get pointwise stability.

    Finally, high order simulations of shock diffracting over a convex wall with two facets were performed. The numerical study is done for a range of Reynolds numbers. By monitoring the velocities at the solid wall, it was shown that the computations were resolved in the boundary layer. Schlieren images from the computational results were obtained which displayed new interesting flow features.

  • 2.
    Abbas, Qaisar
    et al.
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Nordström, Jan
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Weak versus strong no-slip boundary conditions for the Navier-Stokes equations2010In: Engineering Applications of Computational Fluid Mechanics, ISSN 1994-2060, Vol. 4, p. 29-38Article in journal (Refereed)
  • 3.
    Abbas, Qaisar
    et al.
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Nordström, Jan
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Weak versus Strong No-Slip Boundary Conditions for the Navier-Stokes Equations2008In: Proc. 6th South African Conference on Computational and Applied Mechanics, South African Association for Theoretical and Applied Mechanics , 2008, p. 52-62Conference paper (Other academic)
  • 4.
    Abbas, Qaisar
    et al.
    Uppsala universitet, Avdelningen för teknisk databehandling.
    van der Weide, Edwin
    Nordström, Jan
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Accurate and stable calculations involving shocks using a new hybrid scheme2009In: Proc. 19th AIAA CFD Conference, AIAA , 2009Conference paper (Refereed)
  • 5.
    Abbas, Qaisar
    et al.
    Uppsala universitet, Avdelningen för teknisk databehandling.
    van der Weide, Edwin
    Faculty of Engineering Technology, University of Twente, AE Enschede, The Netherlands.
    Nordström, Jan
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Energy Stability of the MUSCL Scheme2010In: Proc. 7th South African Conference on Computational and Applied Mechanics, South African Association for Theoretical and Applied Mechanics , 2010, p. 65:1-8Conference paper (Other academic)
  • 6.
    Abbas, Qaisar
    et al.
    Uppsala universitet, Avdelningen för teknisk databehandling.
    van der Weide, Edwin
    Nordström, Jan
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Energy stability of the MUSCL scheme2010In: Numerical Mathematics and Advanced Applications: 2009, Berlin: Springer-Verlag , 2010, p. 61-68Conference paper (Refereed)
  • 7.
    Amsallem, David
    et al.
    Department of Aeronautics and Astronautics, Stanford University, Stanford, CA 94305-4035, USA.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Energy Stable Model Reduction of Neurons by Non-negative Discrete Empirical Interpolation2016In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 38, no 2, p. B297-B326Article in journal (Refereed)
    Abstract [en]

    The accurate and fast prediction of potential propagation in neuronal networks is of prime importance in neurosciences. This work develops a novel structure-preserving model reduction technique to address this problem based on Galerkin projection and nonnegative operator approximation. It is first shown that the corresponding reduced-order model is guaranteed to be energy stable, thanks to both the structure-preserving approach that constructs a distinct reduced-order basis for each cable in the network and the preservation of nonnegativity. Furthermore, a posteriori error estimates are provided, showing that the model reduction error can be bounded and controlled. Finally, the application to the model reduction of a large-scale neuronal network underlines the capability of the proposed approach to accurately predict the potential propagation in such networks while leading to important speedups.

  • 8.
    Amsallem, David
    et al.
    Department of Aeronautics and Astronautics, Stanford University, Stanford, USA.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    High-order accurate difference schemes for the Hodgkin-Huxley equations2012Report (Other academic)
    Abstract [en]

    A novel approach for simulating potential propagation in neuronal branches with high accuracy is developed. The method relies on high-order accurate dierence schemes using the Summation-By-Parts operators with weak boundary and interface conditions applied to the Hodgkin-Huxley equations. This work is the rst demonstrating high accuracy for that equation. Several boundary conditions are considered including the non-standard one accounting for the soma presence, which is characterized by its own partial dierential equation. Well-posedness for the continuous problem as well as stability of the discrete approximation is proved for all the boundary conditions. Gains in terms of CPU times are observed when high-order operators are used, demonstrating the advantage of the high-order schemes for simulating potential propagation in large neuronal trees.

  • 9.
    Amsallem, David
    et al.
    Department of Aeronautics and Astronautics, Stanford University, USA.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    High-order accurate difference schemes for the Hodgkin-Huxley equations2013In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 252, p. 573-590Article in journal (Refereed)
    Abstract [en]

    A novel approach for simulating potential propagation in neuronal branches with high accuracy is developed. The method relies on high-order accurate difference schemes using the Summation-By-Parts operators with weak boundary and interface conditions applied to the Hodgkin–Huxley equations. This work is the first demonstrating high accuracy for that equation. Several boundary conditions are considered including the non-standard one accounting for the soma presence, which is characterized by its own partial differential equation. Well-posedness for the continuous problem as well as stability of the discrete approximation is proved for all the boundary conditions. Gains in terms of CPU times are observed when high-order operators are used, demonstrating the advantage of the high-order schemes for simulating potential propagation in large neuronal trees.

  • 10.
    Berg, Jens
    et al.
    Uppsala University, Department of Information Technology.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    A stable and dual consistent boundary treatment using finite differences on summation-by-parts form2012In: European Congress on Computational Methods in Applied Sciences and Engineering, Vienna University of Technology , 2012Conference paper (Other academic)
    Abstract [en]

    This paper is concerned with computing very high order accurate linear functionals from a numerical solution of a time-dependent partial differential equation (PDE). Based on finite differences on summation-by-parts form, together with a weak implementation of the boundary conditions, we show how to construct suitable boundary conditions for the PDE such that the continuous problem is well-posed and the discrete problem is stable and spatially dual consistent. These two features result in a superconvergent functional, in the sense that the order of accuracy of the functional is provably higher than that of the solution.

  • 11.
    Berg, Jens
    et al.
    Uppsala University, Sweden.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Duality based boundary conditions and dual consistent finite difference discretizations of the Navier–Stokes and Euler equations2014In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 259, p. 135-153Article in journal (Refereed)
    Abstract [en]

    In this paper we derive new farfield boundary conditions for the time-dependent Navier–Stokes and Euler equations in two space dimensions. The new boundary conditions are derived by simultaneously considering well-posedess of both the primal and dual problems. We moreover require that the boundary conditions for the primal and dual Navier–Stokes equations converge to well-posed boundary conditions for the primal and dual Euler equations.

    We perform computations with a high-order finite difference scheme on summation-by-parts form with the new boundary conditions imposed weakly by the simultaneous approximation term. We prove that the scheme is both energy stable and dual consistent and show numerically that both linear and non-linear integral functionals become superconvergent.

  • 12.
    Berg, Jens
    et al.
    Uppsala University, Department of Information Technology.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Duality based boundary treatment for the Euler and Navier-Stokes equations2013In: AIAA Aerospace Sciences - Fluid Sciences Event, 2013, p. 1-19Conference paper (Other academic)
    Abstract [en]

    In this paper we construct well-posed boundary conditions for the compressible Euler and Navier-Stokes equations in two space dimensions. When also considering the dual equations, we show how to construct the boundary conditions so that both the primal and dual problems are well-posed. By considering the primal and dual problems simultaneously, we construct energy stable and dual consistent finite difference schemes on summation-by-  parts form with weak imposition of the boundary conditions.

    According to linear theory, the stable and dual consistent discretization can be used to compute linear integral functionals from the solution at a superconvergent rate. Here we evaluate numerically the superconvergence property for the non-linear Euler and Navier{ Stokes equations with linear and non-linear integral functionals.

  • 13.
    Berg, Jens
    et al.
    Uppsala University, Department of Information Technology, SE-751 05, Uppsala, Sweden.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    On the impact of boundary conditions on dual consistent finite difference discretizations2013In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 236, p. 41-55Article in journal (Refereed)
    Abstract [en]

    In this paper we derive well-posed boundary conditions for a linear incompletely parabolic system of equations, which can be viewed as a model problem for the compressible Navier{Stokes equations. We show a general procedure for the construction of the boundary conditions such that both the primal and dual equations are wellposed.

    The form of the boundary conditions is chosen such that reduction to rst order form with its complications can be avoided.

    The primal equation is discretized using finite difference operators on summation-by-parts form with weak boundary conditions. It is shown that the discretization can be made energy stable, and that energy stability is sufficient for dual consistency.

    Since reduction to rst order form can be avoided, the discretization is significantly simpler compared to a discretization using Dirichlet boundary conditions.

    We compare the new boundary conditions with standard Dirichlet boundary conditions in terms of rate of convergence, errors and discrete spectra. It is shown that the scheme with the new boundary conditions is not only far simpler, but also has smaller errors, error bounded properties, and highly optimizable eigenvalues, while maintaining all desirable properties of a dual consistent discretization.

  • 14.
    Berg, Jens
    et al.
    Division of Scientific Computing, Department of Information Technology, Uppsala University, Sweden.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Spectral analysis of the continuous and discretized heat and advection equation on single and multiple domains2012In: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 62, no 11, p. 1620-1638Article in journal (Refereed)
    Abstract [en]

    In this paper we study the heat and advectionequation in single and multipledomains. The equations are discretized using a second order accurate finite difference method on Summation-By-Parts form with weak boundary and interface conditions. We derive analytic expressions for the spectrum of the continuous problem and for their corresponding discretization matrices.

    It is shown how the spectrum of the singledomain operator is contained in the multi domain operator spectrum when artificial interfaces are introduced. The interface treatments are posed as a function of one parameter, and the impact on the spectrum and discretization error is investigated as a function of this parameter. Finally we briefly discuss the generalization to higher order accurate schemes.

  • 15.
    Berg, Jens
    et al.
    Uppsala University, Department of Information Technology.
    Nordström, Jan
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.
    Stable Robin solid wall boundary conditions for the Navier-Stokes equations2011In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 230, no 19, p. 7519-7532Article in journal (Refereed)
    Abstract [en]

    In this paper we prove stability of Robin solid wall boundary conditions for the compressible Navier–Stokes equations. Applications include the no-slip boundary conditions with prescribed temperature or temperature gradient and the first order slip-flow boundary conditions. The formulation is uniform and the transitions between different boundary conditions are done by a change of parameters. We give different sharp energy estimates depending on the choice of parameters.

    The discretization is done using finite differences on Summation-By-Parts form with weak boundary conditions using the Simultaneous Approximation Term. We verify convergence by the method of manufactured solutions and show computations of flows ranging from no-slip to almost full slip.

  • 16.
    Berg, Jens
    et al.
    Uppsala University, Department of Information Technology, Sweden.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Superconvergent functional output for time-dependent problems using finite differences on summation-by-parts form2012In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 231, no 20, p. 6846-6860Article in journal (Refereed)
    Abstract [en]

    Finitedifference operators satisfying the summation-by-parts (SBP) rules can be used to obtain high order accurate, energy stable schemes for time-dependent partial differential equations, when the boundary conditions are imposed weakly by the simultaneous approximation term (SAT).

    In general, an SBP-SAT discretization is accurate of order p + 1 with an internal accuracy of 2p and a boundary accuracy of p. Despite this, it is shown in this paper that any linear functional computed from the time-dependent solution, will be accurate of order 2p when the boundary terms are imposed in a stable and dual consistent way.

    The method does not involve the solution of the dual equations, and superconvergent functionals are obtained at no extra computational cost. Four representative model problems are analyzed in terms of convergence and errors, and it is shown in a systematic way how to derive schemes which gives superconvergentfunctionaloutputs.

  • 17. Berggren, Martin
    et al.
    Ekström, Sven-Erik
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Nordström, Jan
    Uppsala universitet, Avdelningen för teknisk databehandling.
    A discontinuous Galerkin extension of the vertex-centered edge-based finite volume method2009In: Communications in Computational Physics, ISSN 1815-2406, Vol. 5, p. 456-468Article in journal (Refereed)
  • 18. Carpenter, Mark H.
    et al.
    Nordström, Jan
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Gottlieb, David
    A stable and conservative interface treatment of arbitrary spatial accuracy1999In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 148, p. 341-365Article in journal (Refereed)
  • 19.
    Carpenter, Mark H
    et al.
    Computational Aerosciences Branch, NASA Langley Research Center, Hampton, VA 23681, USA.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Gottlieb, David
    cDivision of Applied Mathematics, Brown University, Providence, RI 02912, USA.
    Corrigendum to “A stable and conservative interface treatment of arbitrary spatial accuracy” [J.Comput.Phys.148(1999)341–365]2017In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 351, p. 534-Article in journal (Other academic)
  • 20. Carpenter, Mark H.
    et al.
    Nordström, Jan
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Gottlieb, David
    Revisiting and extending interface penalties for multi-domain summation-by-parts operators2009Report (Other academic)
  • 21.
    Carpenter, Mark H.
    et al.
    NASA Langley Research Center.
    Nordström, Jan
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Gottlieb, David
    Brown University.
    Revisiting and Extending Interface Penalties for Multidomain Summation-by-Parts Operators2010In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 45, no 1-3, p. 118-150Article in journal (Refereed)
    Abstract [en]

    A general interface procedure is presented for multi-domain collocation methods satisfying the summation-by-parts (SBP) spatial discretization convention. Unlike more traditional operators (e.g. FEM) applied to the advection-diffusion equation, the new procedure penalizes the solution and the first p derivatives across the interface. The combined interior/interface operators are proven to be pointwise stable, and conservative, although accuracy deteriorates for p≥2. Penalties between two different sets of variables are compared (motivated by FEM primal and flux formulations), and are shown to be equivalent for certain choices of penalty parameters. Extensive validation studies are presented using two classes of high-order SBP operators: (1) central finite difference, and (2) Legendre spectral collocation.

  • 22.
    Delorme, Yann T.
    et al.
    Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa, Israel.
    Puri, Kunal
    Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa, Israel.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Linders, Viktor
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Dong, Suchuan
    Department of Mathematics, Purdue University, West Lafayette, IN, USA.
    Frankel, Steven H.
    Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa, Israel.
    A simple and efficient incompressible Navier-Stokes solver for unsteady complex geometry flows on truncated domains2017In: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 150, p. 84-94Article in journal (Refereed)
    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.

    The full text will be freely available from 2019-04-01 08:00
  • 23.
    Duru, Kenneth
    Uppsala universitet, Avdelningen för beräkningsvetenskap.
    Perfectly Matched Layers and High Order Difference Methods for Wave Equations2012Doctoral thesis, monograph (Other academic)
    Abstract [en]

    The perfectly matched layer (PML) is a novel technique to simulate the absorption of waves in unbounded domains. The underlying equations are often a system of second order hyperbolic partial differential equations. In the numerical treatment, second order systems are often rewritten and solved as first order systems. There are several benefits with solving the equations in second order formulation, though. However, while the theory and numerical methods for first order hyperbolic systems are well developed, numerical techniques to solve second order hyperbolic systems are less complete.

    We construct a strongly well-posed PML for second order systems in two space dimensions, focusing on the equations of linear elasto-dynamics. In the continuous setting, the stability of both first order and second order formulations are linearly equivalent. We have found that if the so-called geometric stability condition is violated, approximating the first order PML with standard central differences leads to a high frequency instability at most resolutions. In the second order setting growth occurs only if growing modes are well resolved. We determine the number of grid points that can be used in the PML to ensure a discretely stable PML, for several anisotropic elastic materials.

    We study the stability of the PML for problems where physical boundaries are important. First, we consider the PML in a waveguide governed by the scalar wave equation. To ensure the accuracy and the stability of the discrete PML, we derived a set of equivalent boundary conditions. Second, we consider the PML for second order symmetric hyperbolic systems on a half-plane. For a class of stable boundary conditions, we derive transformed boundary conditions and prove the stability of the corresponding half-plane problem. Third, we extend the stability analysis to rectangular elastic waveguides, and demonstrate the stability of the discrete PML.

    Building on high order summation-by-parts operators, we derive high order accurate and strictly stable finite difference approximations for second order time-dependent hyperbolic systems on bounded domains. Natural and mixed boundary conditions are imposed weakly using the simultaneous approximation term method. Dirichlet boundary conditions are imposed strongly by injection. By constructing continuous strict energy estimates and analogous discrete strict energy estimates, we prove strict stability.

  • 24. Efraimsson, Gunilla
    et al.
    Forsberg, Nicolas
    Nordström, Jan
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Simulations of acoustic waves in a turbo-fan engine air intake2010In: Proc. 16th AIAA/CEAS Aeroacoustics Conference, AIAA , 2010Conference paper (Refereed)
  • 25. Efraimsson, Gunilla
    et al.
    Gong, Jing
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Svärd, Magnus
    Nordström, Jan
    Uppsala universitet, Avdelningen för teknisk databehandling.
    An investigation of the performance of a high-order accurate Navier-Stokes code2006In: Proc. ECCOMAS CFD Conference 2006, The Netherlands: Tech. Univ. Delft , 2006, p. 11-Conference paper (Refereed)
  • 26. Eliasson, Peter
    et al.
    Eriksson, Sofia
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Nordström, Jan
    Uppsala universitet, Avdelningen för teknisk databehandling.
    The influence of weak and strong solid wall boundary conditions on the convergence to steady-state of the Navier-Stokes equations2009In: Proc. 19th AIAA CFD Conference, AIAA , 2009Conference paper (Refereed)
  • 27.
    Eliasson, Peter
    et al.
    Department of Aeronautics and Autonomous Systems, FOI, Swedish Defense Research Agency, SE-164 90, Stockholm, Sweden .
    Kupiainen, Marco
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Higher Order Accurate Solutions for Flow in a Cavity: Experiences and Lessons Learned2015In: Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014 / [ed] Mejdi Azaïez, Henda El Fekih, Jan S. Hesthaven, Springer, 2015, p. 189-196Chapter in book (Refereed)
    Abstract [en]

    Experiences from using a higher order accurate finite difference multiblock solver to compute the time dependent flow over a cavity is summarized. The work has been carried out as part of a work in a European project called IDIHOM in a collaboration between the Swedish Defense Research Agency (FOI) and University of Linköping (LiU). The higher order code is based on Summation By Parts operators combined with the Simultaneous Approximation Term approach for boundary and interface conditions. The spatial accuracy of the code is verified by calculations over a cyclinder by monitoring the decay of the errors of known wall quantities as the grid is refined. The focus is on the validation for a test case of transonic flow over a rectangular cavity with hybrid RANS/LES calculations. The results are compared to reference numerical results from a second order finite volume code as well as with experimental results with a good overall agreement between the results.

  • 28.
    Eliasson, Peter
    et al.
    FOI, Swedish Defence Research Agency, SE-16490 Stockholm, Sweden.
    Lundquist, Tomas
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    A global time integration approach for realistic unsteady flow computations2016Conference paper (Refereed)
    Abstract [en]

    A novel time integration approach is explored for unsteady flow computations. It is a multi-block formulation in time where one solves for all time levels within a block simultaneously. The time discretization within a block is based on the summation-by-parts (SBP) technique in time combined with the simultaneous-approximation-term (SAT) technique for imposing the initial condition. The approach is implicit, unconditionally stable and can be made high order accurate in time. The implicit system is solved by a dual time stepping technique. The technique has been implemented in a flow solver for unstructured grids and applied to an unsteady flow problem with vortex shedding over a cylinder. Four time integration approaches being 2nd to 5th order accurate in time are evaluated and compared to the conventional 2nd order backward difference (BDF2) method and a 4th order diagonally implicit Runge-Kutta scheme (ESDIRK64). The obtained orders of accuracy are higher than expected and correspond to the accuracy in the interior of the blocks, up to 8th order accuracy is obtained. The influence on the accuracy from the size of the time blocks is small. Smaller blocks are computationally more efficient though, and the efficiency increases with increased accuracy of the SBP operator and reduced size of time steps. The most accurate scheme, with a small time step and block size, is approximately as efficient as the ESDIRK64 scheme. There is a significant potential for improvements ranging from convergence acceleration techniques in dual time, alternative initialization of time blocks, and by introducing smaller time blocks based on alternative SBP operators.

  • 29.
    Eliasson, Peter
    et al.
    Swedish Defence Research Agency (FOI), Stockholm, Sweden.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    The Influence of Viscous Operator and Wall Boundary Conditions on the Accuracy of the Navier-Stokes Equations2013In: AIAA Aerospace Sciences - Fluid Sciences Event, 2013, p. 1-15Conference paper (Other academic)
    Abstract [en]

    The discretization of the viscous operator in an edge-based flow solver for unstructured grids has been investigated. A compact discretization of the Laplace and thin-layer operators in the viscous terms is used with two different wall boundary conditions. Furthermore, a wide discretization of the same operators is investigated. The resulting numerical operators are all formally second order accurate in space; the wide operator has higher truncation errors. The operators are implemented weakly using a penalty formulation and are shown to be stable for a scalar model problem with given constraints on the penalty coefficients. The different operators are applied to a set of grid convergence test cases for laminar flow in two dimensions up to a large-scale three dimensional turbulent flow problem. The operators converge to the same solutions as the grids are refined with one exception where the wide operator converges to a solution with higher drag. The 2nd compact discretization, being locally more accurate at a wall boundary than the original 1st compact operator, reduces the grid dependency somewhat for most test cases. The wide operator performs very well and leads for most test cases to results with minimum spread between coarsest and finest grids. For one test case though, the wide operator has a negative influence on the steady state convergence.

  • 30. Eliasson, Peter
    et al.
    Weinerfelt, Per
    Nordström, Jan
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Application of a line-implicit scheme on stretched unstructured grids2009In: Proc. 47th AIAA Aerospace Sciences Meeting, AIAA , 2009Conference paper (Other academic)
  • 31.
    Erickson, Brittany. A.
    et al.
    Department of Geological Science, San Diego State University, 5500 Campanile Drive, San Diego, California, 92182-1020..
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Stable, High Order Accurate Adaptive Schemes for Long Time, Highly Intermittent Geophysics Problems2013Report (Other academic)
    Abstract [en]

    Many geophysical phenomena are characterized by properties that evolve over a wide range of scales which introduce difficulties when attempting to model these features in one computational method. We have developed a high-order finite difference method for the elastic wave equation that is able to efficiently handle varying temporal scales in a single, stand-alone framework. We apply this method to earthquake cycle models characterized by extremely long interseismic periods interspersed with abrupt, short periods of dynamic rupture. Through the use of summation-by-parts operators and weak enforcement of boundary conditions we derive a provably stable discretization. Time stepping is achieved through the implicit θ-method which allows us to take large time steps during the intermittent period between earthquakes and adapts appropriately to fully resolve rupture.

  • 32.
    Erickson, Brittany A.
    et al.
    Department of Geological Sciences, San Diego State University, San Diego, CA 92182-1020, USA.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Stable, high order accurate adaptive schemes for long time, highly intermittent geophysics problems2014In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 271, p. 328-338Article in journal (Refereed)
    Abstract [en]

    Many geophysical phenomena are characterized by properties that evolve over a wide range of scales which introduce difficulties when attempting to model these features in one computational method. We have developed a high-order finite difference method for the elastic wave equation that is able to efficiently handle varying temporal and spatial scales in a single, stand-alone framework. We apply this method to earthquake cycle models characterized by extremely long interseismic periods interspersed with abrupt, short periods of dynamic rupture. Through the use of summation-by-parts operators and weak enforcement of boundary conditions we derive a provably stable discretization. Time stepping is achieved through the implicit θθ-method which allows us to take large time steps during the intermittent period between earthquakes and adapts appropriately to fully resolve rupture.

  • 33.
    Eriksson, Sofia
    et al.
    Department of Information Technology, Scientific Computing, Uppsala University, SE-751 05 Uppsala, Sweden.
    Abbas, Qaisar
    Department of Information Technology, Scientific Computing, Uppsala University, SE-751 05 Uppsala, Sweden.
    Nordström, Jan
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
    A stable and conservative method for locally adapting the design order of finite difference schemes2011In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 230, no 11, p. 4216-4231Article in journal (Refereed)
    Abstract [en]

    A procedure to locally change the order of accuracy of finite difference schemes is developed. The development is based on existing Summation-By-Parts operators and a weak interface treatment. The resulting scheme is proven to be accurate and stable.

     

    Numerical experiments verify the theoretical accuracy for smooth solutions. In addition, shock calculations are performed, using a scheme where the developed switching procedure is combined with the MUSCL technique.

  • 34.
    Eriksson, Sofia
    et al.
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Abbas, Qaisar
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Nordström, Jan
    Uppsala universitet, Avdelningen för teknisk databehandling.
    A Stable and Conservative Method of Locally Adapting the Design Order of Finite Difference Schemes2010In: Proc. 7th South African Conference on Computational and Applied Mechanics, South African Association for Theoretical and Applied Mechanics , 2010, p. 20:1-9Conference paper (Other academic)
  • 35.
    Eriksson, Sofia
    et al.
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Law, Craig
    Gong, Jing
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Nordström, Jan
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Shock Calculations using a Very High Order Accurate Euler and Navier-Stokes Solver2008In: Proc. 6th South African Conference on Computational and Applied Mechanics, South African Association for Theoretical and Applied Mechanics , 2008, p. 63-73Conference paper (Other academic)
  • 36.
    Eriksson, Sofia
    et al.
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Nordström, Jan
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Analysis of mesh and boundary effects on the accuracy of node-centered finite volume schemes2009In: Proc. 19th AIAA CFD Conference, AIAA , 2009Conference paper (Refereed)
  • 37.
    Eriksson, Sofia
    et al.
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Nordström, Jan
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Analysis of the order of accuracy for node-centered finite volume schemes2009Report (Other academic)
  • 38.
    Eriksson, Sofia
    et al.
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Nordström, Jan
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Analysis of the order of accuracy for node-centered finite volume schemes2009In: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 59, no 10, p. 2659-2676Article in journal (Refereed)
    Abstract [en]

    The order of accuracy of the node-centered finite volume methods is analyzed, and the analysis is based on an exact derivation of the numerical errors in one dimension. The accuracy for various types of grids are considered. Numerical simulations and analysis are performed for both a hyperbolic and a elliptic case, and the results agree. The impact of weakly imposed boundary conditions is analyzed and verified numerically. We show that the error contribution from the primal and dual grid can be treated separately.

  • 39.
    Eriksson, Sofia
    et al.
    Department of Mathematics, Technische Universität Darmstadt, 64293 Darmstadt, Germany.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Exact Non-reflecting Boundary Conditions Revisited: Well-Posedness and Stability2017In: Foundations of Computational Mathematics, ISSN 1615-3375, E-ISSN 1615-3383, Vol. 17, no 4, p. 957-986Article in journal (Refereed)
    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.

  • 40.
    Eriksson, Sofia
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Finite difference schemes with transferable interfaces for parabolic problems2018Report (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.

  • 41.
    Eriksson, Sofia
    et al.
    Department of Information Technology, Uppsala University, Uppsala, Sweden.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Well-posedness and Stability of Exact Non-reflecting Boundary Conditions2013In: AIAA Aerospace Sciences - Fluid Sciences Event, 2013, p. 1-18Conference paper (Other academic)
    Abstract [en]

    Exact non-reflecting boundary conditions for an incompletely parabolic system have been studied. It is shown that well-posedness is a fundamental property of the non-reflecting boundary conditions. By using summation by parts operators for the numerical approximation and a weak boundary implementation, energy stability follows automatically. The stability in combination with the high order accuracy results in a reliable, efficient and accurate method. The theory is supported by numerical simulations.

  • 42.
    Eriksson, Sofia
    et al.
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Svärd, Magnus
    Nordström, Jan
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Simulations of Ground Effects on Wake Vortices at Runways2007Report (Other academic)
  • 43.
    Eriksson, Sofia
    et al.
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Svärd, Magnus
    Nordström, Jan
    Uppsala universitet, Avdelningen för teknisk databehandling.
    Simulations of Ground Effects on Wake Vortices at Runways2008In: Proc. 6th South African Conference on Computational and Applied Mechanics, South African Association for Theoretical and Applied Mechanics , 2008, p. 101-108Conference paper (Refereed)
  • 44.
    Fisher, Travis C.
    et al.
    Computational Aerosciences Branch, NASA Langley Research Center, Hampton, VA 23681, USA.
    Carpenter, Mark H.
    School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Yamaleev, Nail K.
    Department of Mathematics, North Carolina A&T State University, Greensboro, NC 27411, USA.
    Swanson, Charles
    Distinguished Research Associate, NASA Langley Research Center.
    Discretely Conservative Finite-Difference Formulations for Nonlinear Conservation Laws in Split Form: Theory and Boundary Conditions2013In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 234, p. 353-375Article in journal (Refereed)
    Abstract [en]

    The Lax-Wendroff theorem stipulates that a discretely conservative operator is necessary to accurately capture discontinuities. The discrete operator, however, need not be derived from the divergence form of the continuous equations. Indeed, conservation law equations that are split into linear combinations of the divergence and product rule form and then discretized using any diagonal-norm skew-symmetric summation-by-parts (SBP) spatial operator, yield discrete operators that are conservative. Furthermore, split-form, discretely conservation operators can be derived for periodic or finite-domain SBP spatial operators of any order. Examples are presented of a fourth-order, SBP finite-difference operator with second-order boundary closures. Sixth- and eighth-order constructions are derived, and are supplied in an accompanying text file.

  • 45.
    Frenander, Hannes
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    A Provable Stable and Accurate Davies-like Relaxation Procedure Using Multiple Penalty Terms for Lateral Boundaries in Weather Prediction2014Report (Other academic)
    Abstract [en]

    A lateral boundary treatment using summation-by-parts operators and simultaneous approximation terms is introduced. The method, that we refer to as the multiple penalty technique, is similar to Davies relaxation and have similar areas of application. The method is proven, by energy methods, to be stable. We show how to apply this technique on the linearized Euler equations in two space dimensions, and that it reduces the errors in the computational domain.

  • 46.
    Frenander, Hannes
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    A stable and accurate data assmimilation technique using multiple penalty terms in space and time2016Report (Other academic)
    Abstract [en]

    A new method for data assimilation based on weak imposition of external data is introduced. The technique is simple, easy to implement, and the resulting numerical scheme is unconditionally stable. Numerical experiments show that the error growth naturally present in long term simulations can be prevented by using the new technique.

  • 47.
    Frenander, Hannes
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    A Stable and Accurate Davies-like Relaxation Procedure using Multiple Penalty Terms for Lateral Boundary Conditions2016In: Dynamics of atmospheres and oceans (Print), ISSN 0377-0265, E-ISSN 1872-6879, Vol. 73, p. 34-46Article in journal (Refereed)
    Abstract [en]

    A lateral boundary treatment using summation-by-parts operators and simultaneous approximation terms is introduced. The method is similar to Davies relaxation technique used in the weather prediction community and have similar areas of application, but is also provably stable. In this paper, it is shown how this technique can be applied to the shallow water equations, and that it reduces the errors in the computational domain.

  • 48.
    Frenander, Hannes
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    A stable and accurate relaxation technique using multiple penalty terms in space and time2017In: Dynamics of atmospheres and oceans (Print), ISSN 0377-0265, E-ISSN 1872-6879, Vol. 79, p. 10p. 56-65Article in journal (Refereed)
    Abstract [en]

    A new method for data relaxation based on weak imposition of external data is introduced. The technique is simple, easy to implement, and the resulting numerical scheme is unconditionally stable. Numerical experiments show that the error growth naturally present in long term simulations can be prevented by using the new technique.

    The full text will be freely available from 2019-07-29 11:27
  • 49.
    Frenander, Hannes
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Constructing non-reflecting boundary conditions using summation-by-parts in time2016Report (Other academic)
    Abstract [en]

    In this paper we provide a new approach for constructing non-reflecting boundary conditions. The boundary conditions are based on summation-by-parts operators and derived without Laplace transformation in time. We prove that the new non-reflecting boundary conditions yield a well-posed problem and that the corresponding numerical approximation is unconditionally stable. The analysis is demonstrated on a hyperbolic system in two space dimensions, and the theoretical results are confirmed by numerical experiments.

  • 50.
    Frenander, Hannes
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Constructing non-reflecting boundary conditions using summation-by-parts in time2017In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 331, p. 38-48Article in journal (Refereed)
    Abstract [en]

    In this paper we provide a new approach for constructing non-reflecting boundary conditions. The boundary conditions are based on summation-by-parts operators and derived without Laplace transformation in time. We prove that the new non-reflecting boundary conditions yield a well-posed problem and that the corresponding numerical approximation is unconditionally stable. The analysis is demonstrated on a hyperbolic system in two space dimensions, and the theoretical results are confirmed by numerical experiments.

    The full text will be freely available from 2018-11-29 15:37
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