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Winters, Andrew RossORCID iD iconorcid.org/0000-0002-5902-1522
Publications (10 of 36) Show all publications
Ersing, P. & Winters, A. R. (2024). An Entropy Stable Discontinuous Galerkin Method for the Two-Layer Shallow Water Equations on Curvilinear Meshes. Journal of Scientific Computing, 98(3), Article ID 62.
Open this publication in new window or tab >>An Entropy Stable Discontinuous Galerkin Method for the Two-Layer Shallow Water Equations on Curvilinear Meshes
2024 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 98, no 3, article id 62Article in journal (Refereed) Published
Abstract [en]

We present an entropy stable nodal discontinuous Galerkin spectral element method (DGSEM) for the two-layer shallow water equations on two dimensional curvilinear meshes. We mimic the continuous entropy analysis on the semi-discrete level with the DGSEM constructed on Legendre–Gauss–Lobatto (LGL) nodes. The use of LGL nodes endows the collocated nodal DGSEM with the summation-by-parts property that is key in the discrete analysis. The approximation exploits an equivalent flux differencing formulation for the volume contributions, which generate an entropy conservative split-form of the governing equations. A specific combination of a numerical surface flux and discretization of the nonconservative terms is then applied to obtain a high-order path-conservative scheme that is entropy conservative. Furthermore, we find that this combination yields an analogous discretization for the pressure and nonconservative terms such that the numerical method is well-balanced for discontinuous bathymetry on curvilinear domains. Dissipation is added at the interfaces to create an entropy stable approximation that satisfies the second law of thermodynamics in the discrete case, while maintaining the well-balanced property. We conclude with verification of the theoretical findings through numerical tests and demonstrate results about convergence, entropy stability and well-balancedness of the scheme.

Place, publisher, year, edition, pages
Springer, 2024
Keywords
Two-layer shallow water system, Well-balanced method, Discontinuous Galerkin spectral element method, Summation-by-parts, Entropy stability
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-200847 (URN)10.1007/s10915-024-02451-2 (DOI)001158260700001 ()
Funder
Swedish Research Council, 2020-03642VR
Note

Funding: Linköping University; Vetenskapsradet, Sweden; Swedish Research Council [2020-03642 VR];  [2022-06725]

Available from: 2024-02-12 Created: 2024-02-12 Last updated: 2024-02-23
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
Ranocha, H., Schlottke-Lakemper, M., Chan, J., Rueda-Ramírez, A. M., Winters, A. R., Hindenlang, F. & Gassner, G. J. (2023). Efficient implementation of modern entropy stable and kinetic energy preserving discontinuous Galerkin methods for conservation laws. ACM Transactions on Mathematical Software
Open this publication in new window or tab >>Efficient implementation of modern entropy stable and kinetic energy preserving discontinuous Galerkin methods for conservation laws
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2023 (English)In: ACM Transactions on Mathematical Software, ISSN 0098-3500, E-ISSN 1557-7295Article in journal (Refereed) Published
Abstract [en]

Many modern discontinuous Galerkin (DG) methods for conservation laws make use of summation by parts operators and flux differencing to achieve kinetic energy preservation or entropy stability. While these techniques increase the robustness of DG methods significantly, they are also computationally more demanding than standard weak form nodal DG methods. We present several implementation techniques to improve the efficiency of flux differencing DG methods that use tensor product quadrilateral or hexahedral elements, in 2D or 3D respectively. Focus is mostly given to CPUs and DG methods for the compressible Euler equations, although these techniques are generally also useful for other physical systems including the compressible Navier-Stokes and magnetohydrodynamics equations. We present results using two open source codes, Trixi.jl written in Julia and FLUXO written in Fortran, to demonstrate that our proposed implementation techniques are applicable to different code bases and programming languages.

Keywords
flux differencing, entropy stability, conservation laws, summation-by-parts, discontinuous Galerkin
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-198170 (URN)10.1145/3625559 (DOI)
Funder
Swedish Research Council, 2020-03642German Research Foundation (DFG), 2044-39068557German Research Foundation (DFG), 463312734EU, European Research Council, 714487
Available from: 2023-09-27 Created: 2023-09-27 Last updated: 2023-09-27
Ranocha, H., Winters, A. R., Castro, H. G., Dalcin, L., Schlottke-Lakemper, M., Gassner, G. J. & Parsani, M. (2023). On Error-Based Step Size Control for Discontinuous Galerkin Methods for Compressible Fluid Dynamics. Communications on Applied Mathematics and Computation
Open this publication in new window or tab >>On Error-Based Step Size Control for Discontinuous Galerkin Methods for Compressible Fluid Dynamics
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2023 (English)In: Communications on Applied Mathematics and Computation, ISSN 2096-6385Article in journal (Refereed) Epub ahead of print
Abstract [en]

We study a temporal step size control of explicit Runge-Kutta (RK) methods for compressible computational fluid dynamics (CFD), including the Navier-Stokes equations and hyperbolic systems of conservation laws such as the Euler equations. We demonstrate that error-based approaches are convenient in a wide range of applications and compare them to more classical step size control based on a Courant-Friedrichs-Lewy (CFL) number. Our numerical examples show that the error-based step size control is easy to use, robust, and efficient, e.g., for (initial) transient periods, complex geometries, nonlinear shock capturing approaches, and schemes that use nonlinear entropy projections. We demonstrate these properties for problems ranging from well-understood academic test cases to industrially relevant large-scale computations with two disjoint code bases, the open source Julia packages Trixi.jl with OrdinaryDiffEq.jl and the C/Fortran code SSDC based on PETSc.

Place, publisher, year, edition, pages
SPRINGERNATURE, 2023
Keywords
Explicit Runge-Kutta (RK) methods; Step size control; Compressible fluid dynamics; Adaptivity in space and time; Shock capturing
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-194018 (URN)10.1007/s42967-023-00264-y (DOI)000992387000001 ()
Funder
Swedish Research Council, 2020-03642Swedish Research Council, 2018-05973German Research Foundation (DFG), DFG-FOR5409
Note

Funding: Projekt DEAL; Vetenskapsradet, Sweden [2020-03642 VR]; Swedish Research Council [2018-05973]; King Abdullah University of Science and Technology [P2021-0004]; Klaus-Tschira Stiftung via the project "HiFiLab"; Deutsche Forschungsgemeinschaft through the research unit "SNuBIC" [DFG-FOR5409]

Available from: 2023-05-22 Created: 2023-05-22 Last updated: 2023-10-10Bibliographically approved
Nordström, J. & Winters, A. R. (2022). A linear and nonlinear analysis of the shallow water equations and its impact on boundary conditions. Journal of Computational Physics, 463, Article ID 111254.
Open this publication in new window or tab >>A linear and nonlinear analysis of the shallow water equations and its impact on boundary conditions
2022 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 463, article id 111254Article in journal (Refereed) Published
Abstract [en]

We derive boundary conditions and estimates based on the energy and entropy analysis of systems of the nonlinear shallow water equations in two spatial dimensions. It is shown that the energy method provides more details, but is fully consistent with the entropy analysis. The details brought forward by the nonlinear energy analysis allow us to pinpoint where the difference between the linear and nonlinear analysis originate. We find that the result from the linear analysis does not necessarily hold in the nonlinear case. The nonlinear analysis leads in general to a different minimal number of boundary conditions compared with the linear analysis. In particular, and contrary to the linear case, the magnitude of the flow does not influence the number of required boundary conditions.

Place, publisher, year, edition, pages
Elsevier, 2022
Keywords
Energy stability, Entropy stability, Boundary conditions, Nonlinear hyperbolic equations, Shallow water equations
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-184828 (URN)10.1016/j.jcp.2022.111254 (DOI)000831741500007 ()
Note

Funding: Vetenskapsradet, Sweden [2018-05084 VR, 2020-03642 VR]

Available from: 2022-05-09 Created: 2022-05-09 Last updated: 2022-08-18
Iannelli, J., Winters, A. R. & Nordström, J. (2022). ACURA: Acoustics-Convection Upstream Resolution Algorithm. In: AIAA SCITECH 2022 Forum: . Paper presented at AIAA SCITECH 2022 Forum, January 3-7, 2022, San Diego, CA.
Open this publication in new window or tab >>ACURA: Acoustics-Convection Upstream Resolution Algorithm
2022 (English)In: AIAA SCITECH 2022 Forum, 2022Conference paper, Published paper (Refereed)
Abstract [en]

A novel multi-dimensional upwind bias formulation is presented. Designed to solve accurately the Euler and Navier-Stokes conservation-law systems on arbitrary grids, this new formulation is developed within the continuous conservation-law systems ahead of any discretization in space and time. This formulation utilizes shock-jump surface integrals, which emerge within integral weak statements for these conservation laws. The mean value theorem is then applied to express these surface integrals in terms of differences of gradients of differentiable entropy solutions. In the continuum, this process yields shock-capturing entropy weak statements with embedded upwind bias that are consistent with the theory of characteristics. These entropy weak statements are then discretized in space through a Galerkin finite element method, an inherently centered dissipation-free discretization. This automatically generates an intrinsically multi-dimensional upstreamed discrete system on any grid without the need for any further upwinding within the discrete equations. The computational solutions crisply capture shocks and reflect available exact solutions.

National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-182168 (URN)10.2514/6.2022-2589 (DOI)978-1-62410-631-6 (ISBN)
Conference
AIAA SCITECH 2022 Forum, January 3-7, 2022, San Diego, CA
Available from: 2022-01-10 Created: 2022-01-10 Last updated: 2022-01-10
Ranocha, H., Schlottke-Lakemper, M., Winters, A. R., Faulhaber, E., Chan, J. & Gassner, G. J. (2022). Adaptive numerical simulations with Trixi.jl: A case studyof Julia for scientific computing. In: Carsten Bauer (Ed.), JuliaCon Proceedings: . Open Journals, 1
Open this publication in new window or tab >>Adaptive numerical simulations with Trixi.jl: A case studyof Julia for scientific computing
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2022 (English)In: JuliaCon Proceedings / [ed] Carsten Bauer, Open Journals , 2022, Vol. 1, p. 10Conference paper, Published paper (Refereed)
Abstract [en]

We present Trixi.jl, a Julia package for adaptive high-order numericalsimulations of hyperbolic partial differential equations. UtilizingJulia’s strengths, Trixi.jl is extensible, easy to use, and fast.We describe the main design choices that enable these featuresand compare Trixi.jl with a mature open source Fortran code thatuses the same numerical methods.We conclude with an assessmentof Julia for simulation-focused scientific computing, an area thatis still dominated by traditional high-performance computing languagessuch as C, C++, and Fortran.

Place, publisher, year, edition, pages
Open Journals, 2022. p. 10
Keywords
Julia, Scientific Computing, Numerical Simulations, Conservation Laws, Discontinuous Galerkin Methods, Adaptive Mesh Refinement, Compressible Euler Equations, Ideal Magnetohydrodynamics, Entropy Stability, Shock Capturing
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-181964 (URN)10.21105/jcon.00077 (DOI)
Available from: 2021-12-21 Created: 2021-12-21 Last updated: 2022-01-31
Schlottke-Lakemper, M., Winters, A. R., Ranocha, H. & Gassner, G. J. (2021). A purely hyperbolic discontinuous Galerkin approach for self-gravitating gas dynamics. Journal of Computational Physics, 442
Open this publication in new window or tab >>A purely hyperbolic discontinuous Galerkin approach for self-gravitating gas dynamics
2021 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 442Article in journal (Refereed) Published
Abstract [en]

One of the challenges when simulating astrophysical flows with self-gravity is to compute thegravitational forces. In contrast to the hyperbolic hydrodynamic equations, the gravity field isdescribed by an elliptic Poisson equation. We present a purely hyperbolic approach by reformulatingthe elliptic problem into a hyperbolic diffusion problem, which is solved in pseudotime, using the same explicit high-order discontinuous Galerkin method we use for the flow solution. Theflow and the gravity solvers operate on a joint hierarchical Cartesian mesh and are two-way coupledvia the source terms. A key benefit of our approach is that it allows the reuse of existingexplicit hyperbolic solvers without modifications, while retaining their advanced features such asnon-conforming and solution-adaptive grids. By updating the gravitational field in each Runge-Kuttastage of the hydrodynamics solver, high-order convergence is achieved even in coupled multi-physicssimulations. After verifying the expected order of convergence for single-physics and multi-physicssetups, we validate our approach by a simulation of the Jeans gravitational instability.Furthermore, we demonstrate the full capabilities of our numerical framework by computing aself-gravitating Sedov blast with shock capturing in the flow solver and adaptive mesh refinementfor the entire coupled system.

Place, publisher, year, edition, pages
ACADEMIC PRESS INC ELSEVIER SCIENCE, 2021
Keywords
discontinuous Galerkin spectral element method, multi-physics simulation, adaptive mesh refinement, compressible Euler equations, hyperbolic self-gravity
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-176065 (URN)10.1016/j.jcp.2021.110467 (DOI)000671273100002 ()
Funder
Swedish Research Council, 2020-03642EU, European Research Council, 714487German Research Foundation (DFG), 2044-390685587
Note

Funding: European Research Council through the ERC Starting Grant "An Exascale aware and Un-crashable Space-Time-Adaptive Discontinuous Spectral Element Solver for Non-Linear Conservation Laws" [714487]; Vetenskapsradet, SwedenSwedish Research Council [2020-03642 VR]; King Abdullah University of Science and Technology (KAUST)King Abdullah University of Science & Technology; Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germanys Excellence StrategyGerman Research Foundation (DFG) [EXC 2044-390685587]

Available from: 2021-06-03 Created: 2021-06-03 Last updated: 2022-03-17
Rueda-Ramírez, A. M., Hennemann, S., Hindenlang, F., Winters, A. R. & Gassner, G. J. (2021). An Entropy Stable Nodal Discontinuous Galerkin Method for the resistive MHD Equations. Part II: Subcell Finite Volume Shock Capturing. Journal of Computational Physics, 444, Article ID 110580.
Open this publication in new window or tab >>An Entropy Stable Nodal Discontinuous Galerkin Method for the resistive MHD Equations. Part II: Subcell Finite Volume Shock Capturing
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2021 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 444, article id 110580Article in journal (Refereed) Published
Abstract [en]

The second paper of this series presents two robust entropy stable shock-capturing methods for discontinuous Galerkin spectral element (DGSEM) discretizations of the compressible magneto-hydrodynamics (MHD) equations. Specifically, we use the resistive GLM-MHD equations, which include a divergence cleaning mechanism that is based on a generalized Lagrange multiplier (GLM). For the continuous entropy analysis to hold, and due to the divergence-free constraint on the magnetic field, the GLM-MHD system requires the use of non-conservative terms, which need special treatment.

Hennemann et al. ["A provably entropy stable subcell shock capturing approach for high order split form DG for the compressible Euler equations". JCP, 2020] recently presented an entropy stable shock-capturing strategy for DGSEM discretizations of the Euler equations that blends the DGSEM scheme with a subcell first-order finite volume (FV) method. Our first contribution is the extension of the method of Hennemann et al. to systems with non-conservative terms, such as the GLM-MHD equations. In our approach, the advective and non-conservative terms of the equations are discretized with a hybrid FV/DGSEM scheme, whereas the visco-resistive terms are discretized only with the high-order DGSEM method. We prove that the extended method is \change{semi-discretely} entropy stable on three-dimensional unstructured curvilinear meshes. Our second contribution is the derivation and analysis of a second entropy stable shock-capturing method that provides enhanced resolution by using a subcell reconstruction procedure that is carefully built to ensure entropy stability.

We provide a numerical verification of the properties of the hybrid FV/DGSEM schemes on curvilinear meshes and show their robustness and accuracy with common benchmark cases, such as the Orszag-Tang vortex and the GEM (Geospace Environmental Modeling) reconnection challenge. Finally, we simulate a space physics application: the interaction of Jupiter's magnetic field with the plasma torus generated by the moon Io.

Place, publisher, year, edition, pages
ACADEMIC PRESS INC ELSEVIER SCIENCE, 2021
Keywords
Compressible Magnetohydrodynamics, Shock Capturing, Entropy Stability, Discontinuous Galerkin Spectral Element Methods
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-178121 (URN)10.1016/j.jcp.2021.110580 (DOI)000690431700007 ()
Funder
EU, European Research Council, 714487Swedish Research Council, 2020-03642
Note

Funding: European Research CouncilEuropean Research Council (ERC)European Commission [714487]; Vetenskapsradet, SwedenSwedish Research Council [2020-03642 VR]; Regional Computing Center of the University of Cologne (RRZK)

Available from: 2021-08-03 Created: 2021-08-03 Last updated: 2021-09-07
Winters, A. R., Kopriva, D. A., Gassner, G. J. & Hindenlang, F. (2021). Construction of Modern Robust Nodal Discontinuous Galerkin Spectral Element Methods for the Compressible Navier-Stokes Equations (1ed.). In: Martin Kronbichler, Per-Olof Persson (Ed.), Efficient high-order discretizations for computational fluid dynamics: (pp. 117-196). Springer
Open this publication in new window or tab >>Construction of Modern Robust Nodal Discontinuous Galerkin Spectral Element Methods for the Compressible Navier-Stokes Equations
2021 (English)In: Efficient high-order discretizations for computational fluid dynamics / [ed] Martin Kronbichler, Per-Olof Persson, Springer, 2021, 1, p. 117-196Chapter in book (Other academic)
Abstract [en]

Discontinuous Galerkin (DG) methods have a long history in computational physics and engineering to approximate solutions of partial differential equations due to their high-order accuracy and geometric flexibility. However, DG is not perfect and there remain some issues. Concerning robustness, DG has undergone an extensive transformation over the past seven years into its modern form that provides statements on solution boundedness for linear and nonlinear problems. 

This chapter takes a constructive approach to introduce a modern incarnation of the DG spectral element method for the compressible Navier-Stokes equations in a three-dimensional curvilinear context. The groundwork of the numerical scheme comes from classic principles of spectral methods including polynomial approximations and Gauss-type quadratures. We identify aliasing as one underlying cause of the robustness issues for classical DG spectral methods. Removing said aliasing errors requires a particular differentiation matrix and careful discretization of the advective flux terms in the governing equations.

Place, publisher, year, edition, pages
Springer, 2021 Edition: 1
Series
CISM International Centre for Mechanical Sciences ; 602
Keywords
Discontinuous Galerkin methods, Computational physics and engineering
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-176062 (URN)10.1007/978-3-030-60610-7_3 (DOI)9783030606107 (ISBN)
Available from: 2021-06-03 Created: 2021-06-03 Last updated: 2023-02-03Bibliographically approved
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Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0002-5902-1522

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