360 results on '"Jan Nordström"'
Search Results
152. A comparative study of two different shallow water formulations using stable summation by parts schemes
- Author
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S. Hadi Shamsnia, Sarmad Ghader, S. Abbas Haghshenas, and Jan Nordström
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Computational Mathematics ,Applied Mathematics ,Modeling and Simulation ,General Physics and Astronomy - Published
- 2022
153. Corrigendum to 'A stable and conservative interface treatment of arbitrary spatial accuracy' [J. Comput. Phys. 148 (1999) 341-365].
- Author
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Mark H. Carpenter, Jan Nordström, and David I. Gottlieb
- Published
- 2017
- Full Text
- View/download PDF
154. Corrigendum to 'Summation by parts operators for finite difference approximations of second derivatives' [J. Comput. Phys. 199 (2004) 503-540].
- Author
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Ken Mattsson and Jan Nordström
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- 2017
- Full Text
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155. Elimination of First Order Errors in Shock Calculations.
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Gunilla Kreiss, Gunilla Efraimsson, and Jan Nordström
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- 2001
- Full Text
- View/download PDF
156. The Fringe Region Technique and the Fourier Method Used in the Direct Numerical Simulation of Spatially Evolving Viscous Flows.
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Jan Nordström, Niklas Nordin, and Dan S. Henningson
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- 1999
- Full Text
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157. Energy stable wall modeling for the Navier-Stokes equations
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Jan Nordström and Fredrik Laurén
- Subjects
Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Beräkningsmatematik ,Applied Mathematics ,Ill-posed problems ,Computer Science Applications ,Physics::Fluid Dynamics ,Computational Mathematics ,Wall modeling ,Turbulent boundary layer ,Modeling and Simulation ,Navier-Stokes equations ,Penalty procedures ,Stability - Abstract
Close to solid walls, at high Reynolds numbers, fluids may develop steep gradients which require a fine mesh for an accurate simulation of the turbulent boundary layer. An often used cure is to use a wall model instead of a fine mesh, with the drawback that modeling is introduced, leading to possibly unstable numerical schemes. In this paper, we leave the modeling aside, take it for granted, and propose a new set of provably energy stable boundary procedures for the incompressible Navier-Stokes equations. We show that these new boundary procedures lead to numerical results with high accuracy even for coarse meshes where data is partially obtained from a wall model. Funding: Vetenskapsradet, SwedenSwedish Research Council [2018-05084 VR]; Swedish e-Science Research Center (SeRC)
- Published
- 2022
158. Applications of summation-by-parts operators
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Oskar Ålund and Jan Nordström
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Property (philosophy) ,Summation by parts ,Computer science ,Convergence (routing) ,Key (cryptography) ,Stability (learning theory) ,Applied mathematics ,Computational mathematics ,Boundary value problem - Abstract
Numerical solvers of initial boundary value problems will exhibit instabilities and loss of accuracy unless carefully designed. The key property that leads to convergence is stability, which this t ...
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- 2021
159. 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
160. Spectral properties of the incompressible Navier-Stokes equations
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Jan Nordström and Fredrik Laurén
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Physics ,Numerical Analysis ,Steady state ,Physics and Astronomy (miscellaneous) ,Summation by parts ,Discretization ,Beräkningsmatematik ,Applied Mathematics ,Mathematical analysis ,Domain (mathematical analysis) ,Navier Stokes equations ,Computer Science Applications ,Computational Mathematics ,Rate of convergence ,Different boundary condition ,Dispersion relations ,Fourier-Laplace transform ,High-order finite differences ,Incompressible Navier Stokes equations ,Numerical experiments ,Time dependent phenomena ,Modeling and Simulation ,Bounded function ,Decay (organic) ,Laplace transforms ,Viscous flow ,Boundary value problem ,Navier–Stokes equations - Abstract
The influence of different boundary conditions on the spectral properties of the incompressible Navier-Stokes equations is investigated. By using the Fourier-Laplace transform technique, we determine the spectra, extract the decay rate in time, and investigate the dispersion relation. In contrast to an infinite domain, where only diffusion affects the convergence, we show that also the propagation speed influence the rate of convergence to steady state for a bounded domain. Once the continuous equations are analyzed, we discretize using high-order finite-difference operators on summation-by-parts form and demonstrate that the continuous analysis carries over to the discrete setting. The theoretical results are verified by numerical experiments, where we highlight the necessity of high accuracy for a correct description of time-dependent phenomena. Funding agency: The Swedish e-Science Research Centre (SeRC)
- Published
- 2021
161. Stability of Discontinuous Galerkin Spectral Element Schemes for Wave Propagation when the Coefficient Matrices have Jumps
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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
162. Trace preserving quantum dynamics using a novel reparametrization-neutral summation-by-parts difference operator
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Oskar Ålund, Jan Nordström, Takahiro Miura, Fredrik Laurén, Yukinao Akamatsu, and Alexander Rothkopf
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Density matrix ,Trace (linear algebra) ,Dissipative systems ,Physics and Astronomy (miscellaneous) ,Nuclear Theory ,Beräkningsmatematik ,Quantum dynamics ,FOS: Physical sciences ,010103 numerical & computational mathematics ,01 natural sciences ,Initial boundary value problems ,Open quantum systems ,Nuclear Theory (nucl-th) ,High Energy Physics - Phenomenology (hep-ph) ,Master equation ,Time integration ,0101 mathematics ,Physics ,Numerical Analysis ,Summation by parts ,Summation-by-parts operators ,Applied Mathematics ,Operator (physics) ,Computational mathematics ,Computational Physics (physics.comp-ph) ,Mimetic operator ,Computer Science Applications ,010101 applied mathematics ,Computational Mathematics ,High Energy Physics - Phenomenology ,Classical mechanics ,Modeling and Simulation ,Dissipative system ,Physics - Computational Physics - Abstract
We develop a novel numerical scheme for the simulation of dissipative quantum dynamics following from two-body Lindblad master equations. All defining continuum properties of the Lindblad dynamics, hermiticity, positivity and in particular trace conservation of the evolved density matrix are preserved. The central ingredient is a new spatial difference operator, which not only fulfils the summation by parts (SBP) property but also implements a continuum reparametrization property. Using the time evolution of a heavy-quark anti-quark bound state in a hot thermal medium as an explicit example, we show how the reparametrization neutral summation-by-parts (RN-SBP) operator preserves the continuum properties of the theory., 34 pages, 7 figures, open-access code available via https://doi.org/10.5281/zenodo.3744460
- Published
- 2020
163. 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.
- Published
- 2020
164. The relation between primal and dual boundary conditions for hyperbolic systems of equations
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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.
- Published
- 2020
165. Stable and Accurate Filtering Procedures
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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
166. GPU-Acceleration of A High Order Finite Difference Code Using Curvilinear Coordinates
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Jan Nordström, Lilit Axner, Erwin Laure, Marco Kupiainen, and Jing Gong
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High order finite difference method ,Computer science ,Beräkningsmatematik ,Computation ,02 engineering and technology ,GPU programming ,Computational fluid dynamics ,computer.software_genre ,01 natural sciences ,Porting ,010305 fluids & plasmas ,Computational science ,Operator (computer programming) ,0103 physical sciences ,0202 electrical engineering, electronic engineering, information engineering ,020203 distributed computing ,Curvilinear coordinates ,business.industry ,Finite difference ,Solver ,OpenACC ,Computational Mathematics ,Compiler ,General-purpose computing on graphics processing units ,business ,computer - Abstract
GPU-accelerated computing is becoming a popular technology due to the emergence of techniques such as OpenACC, which makes it easy to port codes in their original form to GPU systems using compiler directives, and thereby speeding up computation times relatively simply. In this study we have developed an OpenACC implementation of the high order finite difference CFD solver ESSENSE for simulating compressible flows. The solver is based on summation-by-part form difference operators, and the boundary and interface conditions are weakly implemented using simultaneous approximation terms. This case study focuses on porting code to GPUs for the most time-consuming parts namely sparse matrix vector multiplications and the evaluations of fluxes. The resulting OpenACC implementation is used to simulate the Taylor-Green vortex which produces a maximum speed-up of 61.3 on a single V100 GPU by compared to serial CPU version.
- Published
- 2020
167. Robust boundary conditions for stochastic incompletely parabolic systems of equations
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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
168. 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
169. GPU-acceleration of A High Order Finite Difference Code Using Curvilinear Coordinates
- Author
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Marco, Kupiainen, Gong, Jing, Axner, Lilit, Laure, Erwin, Jan, Nordström, Marco, Kupiainen, Gong, Jing, Axner, Lilit, Laure, Erwin, and Jan, Nordström
- Abstract
GPU-accelerated computing is becoming a popular technology due to the emergence of techniques such as OpenACC, which makes it easy to port codes in their original form to GPU systems using compiler directives, and thereby speeding up computation times relatively simply. In this study we have developed an OpenACC implementation of the high order finite difference CFD solver ESSENSE for simulating compressible flows. The solver is based on summation-by-part form difference operators, and the boundary and interface conditions are weakly implemented using simultaneous approximation terms. This case study focuses on porting code to GPUs for the most time-consuming parts namely sparse matrix vector multiplications and the evaluations of fluxes. The resulting OpenACC implementation is used to simulate the Taylor-Green vortex which produces a maximum speed-up of 61.3 on a single V100 GPU by compared to serial CPU version., QC 20200819
- Published
- 2020
- Full Text
- View/download PDF
170. 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
- Subjects
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
171. 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
172. 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
173. 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
174. 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
175. Reoperations and postoperative complications after osteosynthesis of phalangeal fractures: a retrospective cohort study
- Author
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Jan Nordström, Johanna von Kieseritzky, and Marianne Arner
- Subjects
Adult ,Male ,Reoperation ,medicine.medical_specialty ,Adolescent ,medicine.medical_treatment ,Bone Screws ,030230 surgery ,Finger Phalanges ,Fracture Fixation, Internal ,Fractures, Bone ,Young Adult ,03 medical and health sciences ,Closed Fracture ,Postoperative Complications ,0302 clinical medicine ,medicine ,Humans ,Internal fixation ,Aged ,Retrospective Studies ,Aged, 80 and over ,030222 orthopedics ,Osteosynthesis ,business.industry ,Retrospective cohort study ,Middle Aged ,Phalanx ,Surgery ,Radiography ,Logistic Models ,Female ,business ,Bone Plates ,Bone Wires - Abstract
The aim of the study was to describe the reoperation rates and postoperative complications associated with different methods of osteosynthesis in all extra-articular, closed fractures of the proximal and middle phalanges operated on in the Department of Hand Surgery at Södersjukhuset beween 2010-2014, and to describe the associated patient demographics.This study included all the relevant operations, which comprised operations on 181 fractures in 159 patients (82 male, 77 female), median and mean age = 43 (range = 14-88 years). The clinical records and radiographs were examined retrospectively. A logistic regression analysis was performed on the reoperation data to determine which method of osteosynthesis was the most important descriptor for reoperation, and whether the fracture type was a significant confounder.Forty-seven patients (26%) were reoperated on, mainly due to finger stiffness. The reoperation rates were 25% after K-wire, 15% after screws, and 42% after plate fixation. The unadjusted reoperation rate after plate fixation was significantly higher than for the other methods, but not after adjustment for fracture complexity. The proximal phalanx was affected in 88% of the fractures, and 77% were located in the fourth or fifth finger. Falls, animal-related, and sports injuries were the most frequent causes of injuries.Open reduction with plate fixation was associated with a higher reoperation rate, but this method was also used for the more complex fractures. Plate fixation for phalangeal fractures often entails a need for later tenolysis and plate removal. More aggressive mobilisation regimes might be indicated to prevent adhesion problems.
- Published
- 2017
176. Eigenvalue analysis and convergence acceleration techniques for summation-by-parts approximations
- Author
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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
177. On Stochastic Investigation of Flow Problems Using the Viscous Burgers’ Equation as an Example
- Author
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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
178. Hybrid Computational-Fluid-Dynamics Platform to Investigate Aircraft Trailing Vortices
- Author
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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
179. Galerkin Projection and Numerical Integration for a Stochastic Investigation of the Viscous Burgers’ Equation: An Initial Attempt
- Author
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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
180. 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
181. 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
182. 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
183. Impact of wall modeling on kinetic energy stability for the compressible Navier-Stokes equations
- Author
-
Steven H. Frankel, Vikram Singh, and Jan Nordström
- Subjects
General Computer Science ,FOS: Physical sciences ,Strömningsmekanik och akustik ,Slip (materials science) ,Kinetic energy ,01 natural sciences ,Stability (probability) ,010305 fluids & plasmas ,Physics::Fluid Dynamics ,Stress (mechanics) ,Discontinuous Galerkin method ,0103 physical sciences ,FOS: Mathematics ,Mathematics - Numerical Analysis ,Boundary value problem ,0101 mathematics ,Discontinuous Galerkin ,Skew-symmetric form ,Stability ,Summation-by-parts ,Wall modelling ,Physics ,Fluid Mechanics and Acoustics ,Turbulence ,Fluid Dynamics (physics.flu-dyn) ,General Engineering ,Numerical Analysis (math.NA) ,Physics - Fluid Dynamics ,Mechanics ,Computational Physics (physics.comp-ph) ,010101 applied mathematics ,Norm (mathematics) ,Physics - Computational Physics - Abstract
Affordable, high order simulations of turbulent flows on unstructured grids for very high Reynolds number flows require wall models for efficiency. However, different wall models have different accuracy and stability properties. Here, we develop a kinetic energy stability estimate to investigate stability of wall model boundary conditions. Using this norm, two wall models are studied, a popular equilibrium stress wall model, which is found to be unstable and the dynamic slip wall model which is found to be stable. These results are extended to the discrete case using the Summation by parts (SBP) property of the discontinuous Galerkin method. Numerical tests show that while the equilibrium stress wall model is accurate but unstable, the dynamic slip wall model is inaccurate but stable., Accepted in Computers and Fluids
- Published
- 2021
184. Neural network enhanced computations on coarse grids
- Author
-
Oskar Ålund and Jan Nordström
- Subjects
Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Artificial neural network ,Summation by parts ,Computer science ,Applied Mathematics ,Computation ,010103 numerical & computational mathematics ,Grid ,01 natural sciences ,Computer Science Applications ,Domain (software engineering) ,010101 applied mathematics ,Computational Mathematics ,Modeling and Simulation ,0101 mathematics ,Algorithm - Abstract
Unresolved gradients produce numerical oscillations and inaccurate results. The most straightforward solution to such a problem is to increase the resolution of the computational grid. However, this is often prohibitively expensive and may lead to ecessive execution times. By training a neural network to predict the shape of the solution, we show that it is possible to reduce numerical oscillations and increase both accuracy and efficiency. Data from the neural network prediction is imposed using multiple penalty terms inside the domain.
- Published
- 2021
185. Learning to differentiate
- Author
-
Gianluca Iaccarino, Jan Nordström, and Oskar Ålund
- Subjects
Numerical Analysis ,Theoretical computer science ,Physics and Astronomy (miscellaneous) ,Artificial neural network ,Summation by parts ,Computer science ,Applied Mathematics ,Stability (learning theory) ,010103 numerical & computational mathematics ,Construct (python library) ,Differential operator ,01 natural sciences ,Computer Science Applications ,010101 applied mathematics ,Computational Mathematics ,Modeling and Simulation ,Linear algebra ,Polygon mesh ,0101 mathematics ,Regression algorithm - Abstract
Artificial neural networks together with associated computational libraries provide a powerful framework for constructing both classification and regression algorithms. In this paper we use neural networks to design linear and non-linear discrete differential operators. We show that neural network based operators can be used to construct stable discretizations of initial boundary-value problems by ensuring that the operators satisfy a discrete analogue of integration-by-parts known as summation-by-parts. Our neural network approach with linear activation functions is compared and contrasted with a more traditional linear algebra approach. An application to overlapping grids is explored. The strategy developed in this work opens the door for constructing stable differential operators on general meshes.
- Published
- 2021
186. Theoretical treatment of fluid flow for accelerating bodies
- Author
-
B. W. Skews, Peter Eliasson, Jan Nordström, Irvy Ma Gledhill, H. Roohani, and Karl Forsberg
- Subjects
Angular acceleration ,arbitrary acceleration ,Inertial frame of reference ,Beräkningsmatematik ,Computational Mechanics ,02 engineering and technology ,Computational fluid dynamics ,01 natural sciences ,010305 fluids & plasmas ,Physics::Fluid Dynamics ,fluid physics · Navier-Stokes equations · arbitrary acceleration · manoeuvre · Computational Fluid Dynamics · non-inertial frame ,Acceleration ,0203 mechanical engineering ,0103 physical sciences ,Fluid dynamics ,Navier–Stokes equations ,manoeuvre ,Euler force ,Fluid Flow and Transfer Processes ,Physics ,020301 aerospace & aeronautics ,business.industry ,General Engineering ,Computational Fluid Dynamics ,Mechanics ,Condensed Matter Physics ,Computational Mathematics ,Classical mechanics ,fluid physics ,non-inertial frame ,Navier-Stokes equations ,business ,Non-inertial reference frame - Abstract
Most computational fluid dynamics simulations are, at present, performed in a body-fixed frame, for aeronautical purposes. With the advent of sharp manoeuvre, which may lead to transient effects originating in the acceleration of the centre of mass, there is a need to have a consistent formulation of the Navier–Stokes equations in an arbitrarily moving frame. These expressions should be in a form that allows terms to be transformed between non-inertial and inertial frames and includes gravity, viscous terms, and linear and angular acceleration. Since no effects of body acceleration appear in the inertial frame Navier–Stokes equations themselves, but only in their boundary conditions, it is useful to investigate acceleration source terms in the non-inertial frame. In this paper, a derivation of the energy equation is provided in addition to the continuity and momentum equations previously published. Relevant dimensionless constants are derived which can be used to obtain an indication of the relative significance of acceleration effects. The necessity for using computational fluid dynamics to capture nonlinear effects remains, and various implementation schemes for accelerating bodies are discussed. This theoretical treatment is intended to provide a foundation for interpretation of aerodynamic effects observed in manoeuvre, particularly for accelerating missiles. Funding agencies: DRDB [KT466921, KT528944, KT470887]
- Published
- 2016
187. 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
188. 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
189. 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
190. 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
191. WITHDRAWN: Trace preserving quantum dynamics using a novel reparametrization-neutral summation-by-parts difference operator
- Author
-
Oskar Ålund, Yukinao Akamatsu, Fredrik Laurén, Takahiro Miura, Jan Nordström, and Alexander Rothkopf
- Subjects
Physics and Astronomy (miscellaneous) ,Computer Science Applications - Published
- 2020
192. Level Set Methods for Stochastic Discontinuity Detection in Nonlinear Problems
- Author
-
Jan Nordström, Alireza Doostan, and Per Pettersson
- Subjects
Level set method ,Physics and Astronomy (miscellaneous) ,Beräkningsmatematik ,Computer science ,Hyperbolic PDEs ,MathematicsofComputing_NUMERICALANALYSIS ,Polynomial chaos ,010103 numerical & computational mathematics ,Classification of discontinuities ,01 natural sciences ,Level set methods ,FOS: Mathematics ,Applied mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Uncertainty quantification ,Numerical Analysis ,Level set (data structures) ,Sequence ,Conservation law ,Applied Mathematics ,Numerical Analysis (math.NA) ,Discontinuity tracking ,Computer Science Applications ,010101 applied mathematics ,Computational Mathematics ,Discontinuity (linguistics) ,Modeling and Simulation ,Piecewise - Abstract
Stochastic problems governed by nonlinear conservation laws are challenging due to solution discontinuities in stochastic and physical space. In this paper, we present a level set method to track discontinuities in stochastic space by solving a Hamilton-Jacobi equation. By introducing a speed function that vanishes at discontinuities, the iso-zeros of the level set problem coincide with the discontinuities of the conservation law. The level set problem is solved on a sequence of successively finer grids in stochastic space. The method is adaptive in the sense that costly evaluations of the conservation law of interest are only performed in the vicinity of the discontinuities during the refinement stage. In regions of stochastic space where the solution is smooth, a surrogate method replaces expensive evaluations of the conservation law. The proposed method is tested in conjunction with different sets of localized orthogonal basis functions on simplex elements, as well as frames based on piecewise polynomials conforming to the level set function. The performance of the proposed method is compared to existing adaptive multi-element generalized polynomial chaos methods. Funding agencies: Research Council of Norway through CLIMIT program [244035/E20 CONQUER]; U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research [DE-SC0006402]; NSF [CMMI-1454601]
- Published
- 2018
193. 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
194. Correction: A Stable and Conservative Coupling of the Unsteady Compressible Navier-Stokes Equations at Interfaces Using Finite Difference and Finite Volume Methods
- Author
-
Jan Nordström, Peter Eliasson, and Jing Gong
- Subjects
Physics ,Coupling ,Finite volume method ,Finite difference ,Mechanics ,Compressible navier stokes equations - Published
- 2018
195. A Stable, High Order Accurate and Efficient Hybrid Method for Flow Calculations in Complex Geometries
- Author
-
Jan Nordström and Oskar Ålund
- Subjects
Physics ,Computational Mathematics ,Complex geometry ,Flow (mathematics) ,Discretization ,Beräkningsmatematik ,Mathematical analysis ,Finite difference ,Computational mathematics ,High order ,Domain (software engineering) - Abstract
The suitability of a discretization method is highly dependent on the shape of the domain. Finite difference schemes are typically efficient, but struggle with complex geometry, while finite element methods are expensive but well suited for complex geometries. In this paper we propose a provably stable hybrid method for a 2D advection–diffusion problem, using a class of inner product compatible projection operators to couple the non-conforming grids that arise due to varying the discretization method throughout the domain.
- Published
- 2018
196. 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
197. 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
198. 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
199. 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
200. 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
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