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Linders, V., Carpenter, M. H. & Nordström, J. (2024). A superconvergent stencil-adaptive SBP-SAT finite difference scheme. Journal of Computational Physics, 501, Article ID 112794.
Open this publication in new window or tab >>A superconvergent stencil-adaptive SBP-SAT finite difference scheme
2024 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 501, article id 112794Article in journal (Refereed) Published
Abstract [en]

A stencil-adaptive SBP-SAT finite difference scheme is shown to display superconvergent behavior. As proof of concept, applied to the linear advection equation, it has a convergence rate Ox4) in contrast to a conventional scheme, which converges at a rate Ox3).

Keywords
Summation-by-parts; Adaptivity: Superconvergence
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-200552 (URN)10.1016/j.jcp.2024.112794 (DOI)
Available from: 2024-01-30 Created: 2024-01-30 Last updated: 2024-01-31
Rothkopf, A. & Nordström, J. (2024). A symmetry and Noether charge preserving discretization of initial value problems. Journal of Computational Physics, 498, Article ID 112652.
Open this publication in new window or tab >>A symmetry and Noether charge preserving discretization of initial value problems
2024 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 498, article id 112652Article in journal (Refereed) Published
Abstract [en]

Taking insight from the theory of general relativity, where space and time are treated on the same footing, we develop a novel geometric variational discretization for second order initial value problems (IVPs). By discretizing the dynamics along a world-line parameter, instead of physical time directly, we retain manifest translation symmetry and conservation of the associated continuum Noether charge. A non-equidistant time discretization emerges dynamically, realizing a form of automatic adaptive mesh refinement (AMR), guided by the system symmetries. Using appropriately regularized summation by parts finite difference operators, the continuum Noether charge, defined via the Killing vector associated with translation symmetry, is shown to be exactly preserved in the interior of the simulated time interval. The convergence properties of the approach are demonstrated with two explicit examples.

Place, publisher, year, edition, pages
ACADEMIC PRESS INC ELSEVIER SCIENCE, 2024
Keywords
Initial value problem; Summation by parts; Time-translation invariance; Conserved Noether charge; Adaptive mesh refinement
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-199520 (URN)10.1016/j.jcp.2023.112652 (DOI)001133279600001 ()
Note

Funding: Research Council ofNorway under the FRIPRO Young Research Talent grant [286883]; Swedish Research Council [2021-05484]; UNINETT Sigma2-the National Infrastructure for High Performance Computing and Data Storage in Norway

Available from: 2023-12-08 Created: 2023-12-08 Last updated: 2024-01-23
Lundquist, T., Winters, A. R. & Nordström, J. (2024). Encapsulated generalized summation-by-parts formulations for curvilinear and non-conforming meshes. Journal of Computational Physics, 498, Article ID 112699.
Open this publication in new window or tab >>Encapsulated generalized summation-by-parts formulations for curvilinear and non-conforming meshes
2024 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 498, article id 112699Article in journal (Refereed) Published
Abstract [en]

We extend the construction of so-called encapsulated global summation-by-parts operators to the general case of a mesh which is not boundary conforming. Owing to this development, energy stable discretizations of nonlinear and variable coefficient initial boundary value problems can be formulated in simple and straightforward ways using high-order accurate operators of generalized summation-by-parts type. Encapsulated features on a single computational block or element may include polynomial bases, tensor products as well as curvilinear coordinate transformations. Moreover, through the use of inner product preserving interpolation or projection, the global summation-by-parts property is extended to arbitrary multi-block or multi-element meshes with non-conforming nodal interfaces.

Place, publisher, year, edition, pages
ACADEMIC PRESS INC ELSEVIER SCIENCE, 2024
Keywords
Summation-by-parts; Global difference operators; Curvilinear coordinates; Non-conforming interfaces; Pseudo-spectral methods
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-199530 (URN)10.1016/j.jcp.2023.112699 (DOI)001132589300001 ()
Funder
Swedish Research Council
Available from: 2023-12-11 Created: 2023-12-11 Last updated: 2024-01-23
Glaubitz, J., Nordström, J. & Öffner, P. (2024). Energy-Stable Global Radial Basis Function Methods on Summation-By-Parts Form. Journal of Scientific Computing, 98(1), Article ID 30.
Open this publication in new window or tab >>Energy-Stable Global Radial Basis Function Methods on Summation-By-Parts Form
2024 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 98, no 1, article id 30Article in journal (Refereed) Published
Abstract [en]

Radial basis function methods are powerful tools in numerical analysis and have demonstrated good properties in many different simulations. However, for time-dependent partial differential equations, only a few stability results are known. In particular, if boundary conditions are included, stability issues frequently occur. The question we address in this paper is how provable stability for RBF methods can be obtained. We develop a stability theory for global radial basis function methods using the general framework of summation-by-parts operators often used in the Finite Difference and Finite Element communities. Although we address their practical construction, we restrict the discussion to basic numerical simulations and focus on providing a proof of concept.

Place, publisher, year, edition, pages
SPRINGER/PLENUM PUBLISHERS, 2024
Keywords
Global radial basis functions; Time-dependent partial differential equations; Energy stability; Summation-by-part operators
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-199892 (URN)10.1007/s10915-023-02427-8 (DOI)001137638500001 ()
Note

Funding agencies; Open Access funding enabled and organized by Projekt DEAL. JG was supported by AFOSR #F9550-18-1-0316 and ONR MURI #N00014-20-1-2595. JN was supported by Vetenskapsrådet, Sweden grant 2018-05084 VR and 2021-05484 VR, and the University of Johannesburg. PÖ was supported by the Gutenberg Research College, JGU Mainz.

Available from: 2024-01-03 Created: 2024-01-03 Last updated: 2024-01-24
Nordström, J. (2024). Nonlinear Boundary Conditions for Initial Boundary Value Problems with Applications in Computational Fluid Dynamics. Journal of Computational Physics, 498, Article ID 112685.
Open this publication in new window or tab >>Nonlinear Boundary Conditions for Initial Boundary Value Problems with Applications in Computational Fluid Dynamics
2024 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 498, article id 112685Article in journal (Refereed) Published
Abstract [en]

We derive new boundary conditions and implementation procedures for nonlinear initial boundary value problems (IBVPs) with non-zero boundary data that lead to bounded solutions. The new boundary procedure is applied to nonlinear IBVPs in skew-symmetric form, including dissipative terms. The complete procedure has two main ingredients. Firstly, the energy rate in terms of a surface integral with boundary terms is derived. Secondly, we bound the surface integral by deriving new nonlinear boundary procedures for boundary conditions with non-zero data. The new nonlinear boundary procedure generalises the well known characteristic boundary procedure for linear problems to the nonlinear setting.

To introduce the procedure, a skew-symmetric scalar IBVP encompassing the linear advection equation and Burgers equation is analysed. Once the continuous analysis is done, we show that energy stable nonlinear discrete approximations follow by using summation-by-parts operators combined with weak boundary conditions. The scalar analysis is subsequently repeated for general nonlinear systems of equations. Finally, the new boundary procedure is applied to four important IBVPs in computational fluid dynamics: the incompressible Euler and Navier-Stokes, the shallow water and the compressible Euler equations.

Place, publisher, year, edition, pages
ACADEMIC PRESS INC ELSEVIER SCIENCE, 2024
Keywords
Nonlinear boundary conditions; Navier-Stokes equations; Euler equations; Shallow water equations; Energy and entropy stability; Summation-by-parts
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-199482 (URN)10.1016/j.jcp.2023.112685 (DOI)001133472900001 ()
Funder
Swedish Research Council, 2021-05484 VR
Note

Funding: Vetenskapsradet, Sweden [2021-05484 VR]; University of Johannesburg

Available from: 2023-12-05 Created: 2023-12-05 Last updated: 2024-01-23
Lundquist, T., Malan, A. G. & Nordström, J. (2023). A Method-of-Lines Framework for Energy Stable Arbitrary Lagrangian–Eulerian Methods. SIAM Journal on Numerical Analysis, 61(5), 2327-2351
Open this publication in new window or tab >>A Method-of-Lines Framework for Energy Stable Arbitrary Lagrangian–Eulerian Methods
2023 (English)In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 61, no 5, p. 2327-2351Article in journal (Refereed) Published
Abstract [en]

We present a novel framework based on semi-bounded spatial operators for analyzing and discretizing initial boundary value problems on moving and deforming domains. This development extends an existing framework for well-posed problems and energy stable discretizations from stationary domains to the general case, including arbitrary mesh motion. In particular, we show that an energy estimate derived in the physical coordinate system is equivalent to a semi-bounded property with respect to a stationary reference domain. The continuous analysis leading up to this result is based on a skew-symmetric splitting of the material time derivative and thus relies on the property of integration-by-parts. Following this, a mimetic energy stable arbitrary Lagrangian–Eulerian framework for semi-discretization is formulated, based on approximating the material time derivative in a way consistent with discrete summation-by-parts. Thanks to the semi-bounded property, a method-of-lines approach using standard explicit or implicit time integration schemes can be applied to march the system forward in time. The same type of stability arguments as for the corresponding stationary domain problem applies, without regard to additional properties such as discrete geometric conservation. As an additional bonus we demonstrate that discrete geometric conservation, in the sense of exact free-stream preservation, can still be achieved in an automatic way with the new framework. However, we stress that this is not necessary for stability.

Place, publisher, year, edition, pages
Society for Industrial and Applied Mathematics, 2023
Keywords
moving meshes, energy stability, free-stream preservation, summation-by-parts
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-198702 (URN)10.1137/22m1514945 (DOI)001108608000014 ()
Funder
Swedish Research Council, 2020-03642 VRSwedish Research Council, 2018-05084 VR
Note

Funding: National Research Foundation of South Africa [89916]; Vetenskapsrradet, Sweden [2018-05084 VR]; Swedish e-Science Research Centre (SeRC); SeRC

Available from: 2023-10-23 Created: 2023-10-23 Last updated: 2024-01-23
Rothkopf, A. & Nordström, J. (2023). A new variational discretization technique for initial value problems bypassing governing equations. Journal of Computational Physics, 477, Article ID 111942.
Open this publication in new window or tab >>A new variational discretization technique for initial value problems bypassing governing equations
2023 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 477, article id 111942Article in journal (Refereed) Published
Abstract [en]

Motivated by the fact that both the classical and quantum description of nature rest on causality and a variational principle, we develop a novel and highly versatile discretization prescription for classical initial value problems (IVPs). It is based on an optimization (action) functional with doubled degrees of freedom, which is discretized using a single regularized summation-by-parts (SBP) operator. Formulated as optimization task it allows us to obtain classical trajectories without the need to derive an equation of motion. The novel regularization we develop in this context is inspired by the weak imposition of initial data, often deployed in the modern treatment of IVPs and is implemented using affine coordinates. We demonstrate numerically the stability, accuracy and convergence properties of our approach in systems with classical equations of motion featuring both first and second order derivatives in time.

Place, publisher, year, edition, pages
Elsevier, 2023
Keywords
Initial value problem, Summation by parts, Variational principle
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-191551 (URN)10.1016/j.jcp.2023.111942 (DOI)000927295300001 ()
Note

Funding: Research Council of Norway under the FRIPRO Young Research Talent [286883]; Swedish Research Council [2018-05084, 2021-05484]

Available from: 2023-01-31 Created: 2023-01-31 Last updated: 2023-03-15
Nchupang, M. P., Malan, A. G., Laurén, F. & Nordström, J. (2023). A provably stable and high-order accurate finite difference approximation for the incompressible boundary layer equations. Computers & Fluids, 267, Article ID 106073.
Open this publication in new window or tab >>A provably stable and high-order accurate finite difference approximation for the incompressible boundary layer equations
2023 (English)In: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 267, article id 106073Article in journal (Refereed) Published
Abstract [en]

In this article we develop a high order accurate method to solve the incompressible boundary layer equations in a provably stable manner. We first derive continuous energy estimates, and then proceed to the discrete setting. We formulate the discrete approximation using high-order finite difference methods on summation-by-parts form and implement the boundary conditions weakly using the simultaneous approximation term method. By applying the discrete energy method and imitating the continuous analysis, the discrete estimate that resembles the continuous counterpart is obtained proving stability. We also show that these newly derived boundary conditions removes the singularities associated with the null-space of the nonlinear discrete spatial operator. Numerical experiments that verifies the high-order accuracy of the scheme and coincides with the theoretical results are presented. The numerical results are compared with the well-known Blasius similarity solution as well as that resulting from the solution of the incompressible Navier–Stokes equations.

Place, publisher, year, edition, pages
PERGAMON-ELSEVIER SCIENCE LTD, 2023
Keywords
Incompressible Navier–Stokes equationsBoundary layer equationsHigh order methods; Summation-by-parts; Boundary conditions; Simultaneous approximation terms
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-198435 (URN)10.1016/j.compfluid.2023.106073 (DOI)001098024100001 ()
Funder
Swedish Research Council, 2018-05084 VREU, Horizon 2020, 815044
Note

Funding: National Research Foundation of South Africa [89916]; European Union [815044]; Vetenskapsradet, Sweden [2018-05084 VR, 2021-0584]

Available from: 2023-10-12 Created: 2023-10-12 Last updated: 2023-12-06
Abgrall, R., Nordström, J., Öffner, P. & Tokareva, S. (2023). Analysis of the SBP-SAT Stabilization for Finite Element Methods Part II: Entropy Stability. Communications on Applied Mathematics and Computation, 5(2), 573-595
Open this publication in new window or tab >>Analysis of the SBP-SAT Stabilization for Finite Element Methods Part II: Entropy Stability
2023 (English)In: Communications on Applied Mathematics and Computation, ISSN 2096-6385, Vol. 5, no 2, p. 573-595Article in journal (Refereed) Published
Abstract [en]

In the hyperbolic research community, there exists the strong belief that a continuous Galerkin scheme is notoriously unstable and additional stabilization terms have to be added to guarantee stability. In the first part of the series [6], the application of simultaneous approximation terms for linear problems is investigated where the boundary conditions are imposed weakly. By applying this technique, the authors demonstrate that a pure continuous Galerkin scheme is indeed linearly stable if the boundary conditions are imposed in the correct way. In this work, we extend this investigation to the nonlinear case and focus on entropy conservation. By switching to entropy variables, we provide an estimation of the boundary operators also for nonlinear problems, that guarantee conservation. In numerical simulations, we verify our theoretical analysis.

Place, publisher, year, edition, pages
Springer, 2023
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-193556 (URN)10.1007/s42967-020-00086-2 (DOI)000884643900001 ()2-s2.0-8513239947 (Scopus ID)
Available from: 2023-05-05 Created: 2023-05-05 Last updated: 2023-05-05
Glaubitz, J., Klein, S.-C., Nordström, J. & Öffner, P. (2023). Multi-dimensional summation-by-parts operators for general function spaces: Theory and construction. Journal of Computational Physics, 491, Article ID 112370.
Open this publication in new window or tab >>Multi-dimensional summation-by-parts operators for general function spaces: Theory and construction
2023 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 491, article id 112370Article in journal (Refereed) Published
Abstract [en]

Summation-by-parts (SBP) operators allow us to systematically develop energy-stable and high-order accurate numerical methods for time-dependent differential equations. Until recently, the main idea behind existing SBP operators was that polynomials can accurately approximate the solution, and SBP operators should thus be exact for them. However, polynomials do not provide the best approximation for some problems, with other approximation spaces being more appropriate. We recently addressed this issue and developed a theory for one-dimensional SBP operators based on general function spaces, coined function-space SBP (FSBP) operators. In this paper, we extend the theory of FSBP operators to multiple dimensions. We focus on their existence, connection to quadratures, construction, and mimetic properties. A more exhaustive numerical demonstration of multi-dimensional FSBP (MFSBP) operators and their application will be provided in future works. Similar to the one-dimensional case, we demonstrate that most of the established results for polynomial-based multi-dimensional SBP (MSBP) operators carry over to the more general class of MFSBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than is currently done. This can increase the accuracy of the numerical solutions and/or provide stability to the methods. © 2023 The Author(s)

Place, publisher, year, edition, pages
Elsevier, 2023
Keywords
General function spaces, Initial boundary value problems, Mimetic discretization, Multi-dimensional, Stability, Summation-by-parts operators
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-196538 (URN)10.1016/j.jcp.2023.112370 (DOI)001052590900001 ()2-s2.0-85166326115 (Scopus ID)
Note

Funding agencies:AFOSR #F9550-18-1-0316, the US DOD (ONR MURI) grant #N00014-20-1-2595, the US DOE (SciDAC program) grant #DE-SC0012704, Vetenskapsrådet Sweden grant 2018-05084 VR and 2021-05484, the Swedish e-Science Research Center (SeRC), and the Gutenberg Research College, JGU Mainz. “Oberwolfach Research Fellows” (#2235p) by the Mathematisches Forschungsinstitut Oberwolfach in 2022.

Available from: 2023-08-10 Created: 2023-08-10 Last updated: 2023-09-13
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ORCID iD: ORCID iD iconorcid.org/0000-0002-7972-6183

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