liu.seSearch for publications in DiVA
Change search
Link to record
Permanent link

Direct link
BETA
Winters, Andrew RossORCID iD iconorcid.org/0000-0002-5902-1522
Publications (10 of 24) Show all publications
Nordström, J. & Winters, A. R. (2019). Energy versus entropy estimates for nonlinear hyperbolic systems of equations. Linköping: Linköping University Electronic Press
Open this publication in new window or tab >>Energy versus entropy estimates for nonlinear hyperbolic systems of equations
2019 (English)Report (Other academic)
Abstract [en]

We compare and contrast information provided by the energy analysis of Kreiss and the entropy theory of Tadmor for systems of nonlinear hyperbolic conservation laws. The two-dimensional nonlinear shallow water equations are used to highlight the similarities and differences since the total energy of the system is a mathematical entropy function. We demonstrate that the classical energy method is consistent with the entropy analysis, but significantly more fundamental as it guides proper boundary treatments. In particular, the energy analysis provides information on what type of and how many boundary conditions are required, which is lacking in the entropy analysis. For the shallow water system we determine the number and the type of boundary conditions needed for subcritical and supercritical flows on a general domain. As eigenvalues are augmented in the nonlinear analysis, we find that a flow may be classified as subcritical, but the treatment of the boundary resembles that of a supercritical flow. Because of this, we show that the nonlinear energy analysis leads to a different number of boundary conditions compared with the linear energy analysis. We also demonstrate that the entropy estimate leads to erroneous boundary treatments by over specifying and/or under specifying boundary data causing the loss of existence and/or energy bound, respectively. Our analysis reveals that the nonlinear energy analysis is the only one that provides an estimate for open boundaries. Both the entropy and linear energy analysis fail.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2019. p. 23
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2019:8
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-159114 (URN)LiTH-MAT-R--2019/08--SE (ISRN)
Available from: 2019-07-29 Created: 2019-07-29 Last updated: 2019-08-13
Friedrich, L., Schnücke, G., Winters, A. R., Fernández, D. C., Gassner, G. J. & Carpenter, M. H. (2019). Entropy Stable Space-Time Discontinuous Galerkin Schemes with Summation-by-Parts Property for Hyperbolic Conservation Laws. Journal of Scientific Computing, 80(1), 175-222
Open this publication in new window or tab >>Entropy Stable Space-Time Discontinuous Galerkin Schemes with Summation-by-Parts Property for Hyperbolic Conservation Laws
Show others...
2019 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 80, no 1, p. 175-222Article in journal (Refereed) Published
Abstract [en]

This work examines the development of an entropy conservative (for smooth solutions) or entropy stable (for discontinuous solutions) space-time discontinuous Galerkin (DG) method for systems of nonlinear hyperbolic conservation laws. The resulting numerical scheme is fully discrete and provides a bound on the mathematical entropy at any time according to its initial condition and boundary conditions. The crux of the method is that discrete derivative approximations in space and time are summation-by-parts (SBP) operators. This allows the discrete method to mimic results from the continuous entropy analysis and ensures that the complete numerical scheme obeys the second law of thermodynamics. Importantly, the novel method described herein does not assume any exactness of quadrature in the variational forms that naturally arise in the context of DG methods. Typically, the development of entropy stable schemes is done on the semidiscrete level ignoring the temporal dependence. In this work, we demonstrate that creating an entropy stable DG method in time is similar to the spatial discrete entropy analysis, but there are important (and subtle) differences. Therefore, we highlight the temporal entropy analysis throughout this work. For the compressible Euler equations, the preservation of kinetic energy is of interest besides entropy stability. The construction of kinetic energy preserving (KEP) schemes is, again, typically done on the semidiscrete level similar to the construction of entropy stable schemes. We present a generalization of the KEP condition from Jameson to the space-time framework and provide the temporal components for both entropy stability and kinetic energy preservation. The properties of the space-time DG method derived herein are validated through numerical tests for the compressible Euler equations. Additionally, we provide, in appendices, how to construct the temporal entropy stable components for the shallow water or ideal magnetohydrodynamic (MHD) equations.

Place, publisher, year, edition, pages
Springer, 2019
Keywords
Space-Time Discontinuous Galerkin, Summation-by-Parts, Entropy Conservation, Entropy Stability, Kinetic Energy Preservation
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-156874 (URN)10.1007/s10915-019-00933-2 (DOI)2-s2.0-85062733957 (Scopus ID)
Funder
EU, European Research Council, 714487
Available from: 2019-05-14 Created: 2019-05-14 Last updated: 2019-05-24Bibliographically approved
Bohm, M., Schermeng, S., Winters, A. R., Gassner, G. J. & Jacobs, G. B. (2019). Multi-element SIAC Filter for Shock Capturing Applied to High-Order Discontinuous Galerkin Spectral Element Methods. Journal of Scientific Computing, 81(2), 820-844
Open this publication in new window or tab >>Multi-element SIAC Filter for Shock Capturing Applied to High-Order Discontinuous Galerkin Spectral Element Methods
Show others...
2019 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, ISSN 0885-7474, Vol. 81, no 2, p. 820-844Article in journal (Refereed) Published
Abstract [en]

We build a multi-element variant of the smoothness increasing accuracy conserving (SIAC) shock capturing technique proposed for single element spectral methods by Wissink et al. (J Sci Comput 77:579–596, 2018). In particular, the baseline scheme of our method is the nodal discontinuous Galerkin spectral element method (DGSEM) for approximating the solution of systems of conservation laws. It is well known that high-order methods generate spurious oscillations near discontinuities which can develop in the solution for nonlinear problems, even when the initial data is smooth. We propose a novel multi-element SIAC filtering technique applied to the DGSEM as a shock capturing method. We design the SIAC filtering such that the numerical scheme remains high-order accurate and that the shock capturing is applied adaptively throughout the domain. The shock capturing method is derived for general systems of conservation laws. We apply the novel SIAC filter to the two-dimensional Euler and ideal magnetohydrodynamics equations to several standard test problems with a variety of boundary conditions.

Place, publisher, year, edition, pages
Springer-Verlag New York, 2019
Keywords
Discontinuous Galerkin, Nonlinear hyperbolic conservation laws, SIAC filtering, Shock capturing
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-160123 (URN)10.1007/s10915-019-01036-8 (DOI)000491440200008 ()
Funder
EU, European Research Council, 714487
Note

Funding agencies:  Linkoping University; European Research Council (ERC) under the European Unions Eights Framework Program Horizon 2020 with the research project Extreme, ERCEuropean Research Council (ERC) [714487]; NSF-DMSNational Science Foundation (NSF) [1115705]

Available from: 2019-09-06 Created: 2019-09-06 Last updated: 2019-11-27Bibliographically approved
Winters, A. R., Moura, R. C., Mengaldo, G., Gassner, G. J., Walch, S., Peiro, J. & Sherwin, S. J. (2018). A comparative study on polynomial dealiasing and split form discontinuous Galerkin schemes for under-resolved turbulence computations. Journal of Computational Physics, 372, 1-21
Open this publication in new window or tab >>A comparative study on polynomial dealiasing and split form discontinuous Galerkin schemes for under-resolved turbulence computations
Show others...
2018 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 372, p. 1-21Article in journal (Refereed) Published
Abstract [en]

This work focuses on the accuracy and stability of high-order nodal discontinuous Galerkin (DG) methods for under-resolved turbulence computations. In particular we consider the inviscid Taylor-Green vortex (TGV) flow to analyse the implicit large eddy simulation (iLES) capabilities of DG methods at very high Reynolds numbers. The governing equations are discretised in two ways in order to suppress aliasing errors introduced into the discrete variational forms due to the under-integration of non-linear terms. The first, more straightforward way relies on consistent/over-integration, where quadrature accuracy is improved by using a larger number of integration points, consistent with the degree of the non-linearities. The second strategy, originally applied in the high-order finite difference community, relies on a split (or skew-symmetric) form of the governing equations. Different split forms are available depending on how the variables in the non-linear terms are grouped. The desired split form is then built by averaging conservative and non-conservative forms of the governing equations, although conservativity of the DG scheme is fully preserved. A preliminary analysis based on Burgers’ turbulence in one spatial dimension is conducted and shows the potential of split forms in keeping the energy of higher-order polynomial modes close to the expected levels. This indicates that the favourable dealiasing properties observed from split-form approaches in more classical schemes seem to hold for DG. The remainder of the study considers a comprehensive set of (under-resolved) computations of the inviscid TGV flow and compares the accuracy and robustness of consistent/over-integration and split form discretisations based on the local Lax-Friedrichs and Roe-type Riemann solvers. Recent works showed that relevant split forms can stabilize higher-order inviscid TGV test cases otherwise unstable even with consistent integration. Here we show that stable high-order cases achievable with both strategies have comparable accuracy, further supporting the good dealiasing properties of split form DG. The higher-order cases achieved only with split form schemes also displayed all the main features expected from consistent/over-integration. Among test cases with the same number of degrees of freedom, best solution quality is obtained with Roe-type fluxes at moderately high orders (around sixth order). Solutions obtained with very high polynomial orders displayed spurious features attributed to a sharper dissipation in wavenumber space. Accuracy differences between the two dealiasing strategies considered were, however, observed for the low-order cases, which also yielded reduced solution quality compared to high-order results.

Place, publisher, year, edition, pages
Elsevier, 2018
Keywords
Spectral element methods, Discontinuous Galerkin, Polynomial dealiasing, Split form schemes, Implicit large eddy simulation, Inviscid Taylor-Green vortex
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-156849 (URN)10.1016/j.jcp.2018.06.016 (DOI)000443284400001 ()2-s2.0-85048835225 (Scopus ID)
Funder
EU, European Research Council, 714487German Research Foundation (DFG), SPP 1573EU, European Research Council, 679852
Available from: 2019-05-14 Created: 2019-05-14 Last updated: 2019-05-23Bibliographically approved
Wintermeyer, N., Winters, A. R., Gassner, G. J. & Warburton, T. (2018). An entropy stable discontinuous Galerkin method for the shallow water equations on curvilinear meshes with wet/dry fronts accelerated by GPUs. Journal of Computational Physics, 375, 447-480
Open this publication in new window or tab >>An entropy stable discontinuous Galerkin method for the shallow water equations on curvilinear meshes with wet/dry fronts accelerated by GPUs
2018 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 375, p. 447-480Article in journal (Refereed) Published
Abstract [en]

We extend the entropy stable high order nodal discontinuous Galerkin spectral element approximation for the non-linear two dimensional shallow water equations presented by Wintermeyer et al. [N. Wintermeyer, A. R. Winters, G. J. Gassner, and D. A. Kopriva. An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry. Journal of Computational Physics, 340:200-242, 2017] with a shock capturing technique and a positivity preservation capability to handle dry areas. The scheme preserves the entropy inequality, is well-balanced and works on unstructured, possibly curved, quadrilateral meshes. For the shock capturing, we introduce an artificial viscosity to the equations and prove that the numerical scheme remains entropy stable. We add a positivity preserving limiter to guarantee non-negative water heights as long as the mean water height is non-negative. We prove that non-negative mean water heights are guaranteed under a certain additional time step restriction for the entropy stable numerical interface flux. We implement the method on GPU architectures using the abstract language OCCA, a unified approach to multi-threading languages. We show that the entropy stable scheme is well suited to GPUs as the necessary extra calculations do not negatively impact the runtime up to reasonably high polynomial degrees (around N = 7). We provide numerical examples that challenge the shock capturing and positivity properties of our scheme to verify our theoretical findings.

Place, publisher, year, edition, pages
Elsevier, 2018
Keywords
Shallow water equations, Discontinuous Galerkin spectral element method, Shock capturing, Positivity preservation, GPUs, OCCA
National Category
Computational Mathematics Oceanography, Hydrology and Water Resources
Identifiers
urn:nbn:se:liu:diva-156850 (URN)10.1016/j.jcp.2018.08.038 (DOI)000450907600022 ()2-s2.0-85052988975 (Scopus ID)
Available from: 2019-05-14 Created: 2019-05-14 Last updated: 2019-05-23Bibliographically approved
Friedrich, L., Winters, A. R., Fernández, D. C., Gassner, G. J., Parsani, M. & Carpenter, M. H. (2018). An Entropy Stable h/p Non-Conforming Discontinuous Galerkin Method with the Summation-by-Parts Property. Journal of Scientific Computing, 77(2), 689-725
Open this publication in new window or tab >>An Entropy Stable h/p Non-Conforming Discontinuous Galerkin Method with the Summation-by-Parts Property
Show others...
2018 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 77, no 2, p. 689-725Article in journal (Refereed) Published
Abstract [en]

This work presents an entropy stable discontinuous Galerkin (DG) spectral element approximation for systems of non-linear conservation laws with general geometric (h) and polynomial order (p) non-conforming rectangular meshes. The crux of the proofs presented is that the nodal DG method is constructed with the collocated Legendre–Gauss–Lobatto nodes. This choice ensures that the derivative/mass matrix pair is a summation-by-parts (SBP) operator such that entropy stability proofs from the continuous analysis are discretely mimicked. Special attention is given to the coupling between non-conforming elements as we demonstrate that the standard mortar approach for DG methods does not guarantee entropy stability for non-linear problems, which can lead to instabilities. As such, we describe a precise procedure and modify the mortar method to guarantee entropy stability for general non-linear hyperbolic systems on h / p non-conforming meshes. We verify the high-order accuracy and the entropy conservation/stability of fully non-conforming approximation with numerical examples.

Place, publisher, year, edition, pages
Springer, 2018
Keywords
Summation-by-Parts, Discontinuous Galerkin, Entropy Conservation, Entropy Stability, h/p Non-Conforming Mesh, Non-Linear Hyperbolic Conservation Laws
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-156856 (URN)10.1007/s10915-018-0733-7 (DOI)000446594600001 ()2-s2.0-85046752892 (Scopus ID)
Funder
German Research Foundation (DFG), TA 1260/1-1EU, European Research Council, 714487
Available from: 2019-05-14 Created: 2019-05-14 Last updated: 2019-05-23Bibliographically approved
Bohm, M., Winters, A. R., Gassner, G. J., Derigs, D., Hindenlang, F. & Saur, J. (2018). An entropy stable nodal discontinuous Galerkin method for the resistive MHD equations. Part I: Theory and numerical verification. Journal of Computational Physics
Open this publication in new window or tab >>An entropy stable nodal discontinuous Galerkin method for the resistive MHD equations. Part I: Theory and numerical verification
Show others...
2018 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716Article in journal (Refereed) In press
Abstract [en]

The first paper of this series presents a discretely entropy stable discontinuous Galerkin (DG) method for the resistive magnetohydrodynamics (MHD) equations on three-dimensional curvilinear unstructured hexahedral meshes. Compared to other fluid dynamics systems such as the shallow water equations or the compressible Navier-Stokes equations, the resistive MHD equations need special considerations because of the divergence-free constraint on the magnetic field. For instance, it is well known that for the symmetrization of the ideal MHD system as well as the continuous entropy analysis a non-conservative term proportional to the divergence of the magnetic field, typically referred to as the Powell term, must be included. As a consequence, the mimicry of the continuous entropy analysis in the discrete sense demands a suitable DG approximation of the non-conservative terms in addition to the ideal MHD terms.

This paper focuses on the resistive MHD equations: Our first contribution is a proof that the resistive terms are symmetric and positive-definite when formulated in entropy space as gradients of the entropy variables, which enables us to show that the entropy inequality holds for the resistive MHD equations. This continuous analysis is the key for our DG discretization and guides the path for the construction of an approximation that discretely mimics the entropy inequality, typically termed entropy stability. Our second contribution is a detailed derivation and analysis of the discretization on three-dimensional curvilinear meshes. The discrete analysis relies on the summation-by-parts property, which is satisfied by the DG spectral element method (DGSEM) with Legendre-Gauss-Lobatto (LGL) nodes. Although the divergence-free constraint is included in the non-conservative terms, the resulting method has no particular treatment of the magnetic field divergence errors, which might pollute the solution quality. Our final contribution is the extension of the standard resistive MHD equations and our DG approximation with a divergence cleaning mechanism that is based on a generalized Lagrange multiplier (GLM).

As a conclusion to the first part of this series, we provide detailed numerical validations of our DGSEM method that underline our theoretical derivations. In addition, we show a numerical example where the entropy stable DGSEM demonstrates increased robustness compared to the standard DGSEM.

Place, publisher, year, edition, pages
Elsevier, 2018
Keywords
resistive magnetohydrodynamics, entropy stability, discontinuous Galerkin spectral element method, hyperbolic divergence cleaning, curvilinear hexahedral mesh, summation-by- parts
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-156851 (URN)10.1016/j.jcp.2018.06.027 (DOI)2-s2.0-85049958774 (Scopus ID)
Funder
EU, European Research Council, 714487
Available from: 2019-05-14 Created: 2019-05-14 Last updated: 2019-05-23Bibliographically approved
Friedrich, L., Fernández, D. C., Winters, A. R., Gassner, G. J., Zingg, D. W. & Hicken, J. (2018). Conservative and stable degree preserving SBP operators for non-conforming meshes. Journal of Scientific Computing, 75(2), 657-686
Open this publication in new window or tab >>Conservative and stable degree preserving SBP operators for non-conforming meshes
Show others...
2018 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 75, no 2, p. 657-686Article in journal (Refereed) Published
Abstract [en]

Non-conforming numerical approximations offer increased flexibility for applications that require high resolution in a localized area of the computational domain or near complex geometries. Two key properties for non-conforming methods to be applicable to real world applications are conservation and energy stability. The summation-by-parts (SBP) property, which certain finite-difference and discontinuous Galerkin methods have, finds success for the numerical approximation of hyperbolic conservation laws, because the proofs of energy stability and conservation can discretely mimic the continuous analysis of partial differential equations. In addition, SBP methods can be developed with high-order accuracy, which is useful for simulations that contain multiple spatial and temporal scales. However, existing non-conforming SBP schemes result in a reduction of the overall degree of the scheme, which leads to a reduction in the order of the solution error. This loss of degree is due to the particular interface coupling through a simultaneous-approximation-term (SAT). We present in this work a novel class of SBP-SAT operators that maintain conservation, energy stability, and have no loss of the degree of the scheme for non-conforming approximations. The new degree preserving discretizations require an ansatz that the norm matrix of the SBP operator is of a degree ≥ 2p, in contrast to, for example, existing finite difference SBP operators, where the norm matrix is 2p − 1 accurate. We demonstrate the fundamental properties of the new scheme with rigorous mathematical analysis as well as numerical verification.

Place, publisher, year, edition, pages
Springer-Verlag New York, 2018
Keywords
First derivative, Summation-by-parts, Simultaneous-approximation-term, Conservation, Energy stability, Finite difference methods, Non-conforming methods, Intermediate grids
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-156858 (URN)10.1007/s10915-017-0563-z (DOI)000428565100004 ()2-s2.0-85030107035 (Scopus ID)
Funder
German Research Foundation (DFG), TA 2160/1-1
Available from: 2019-05-14 Created: 2019-05-14 Last updated: 2019-05-24Bibliographically approved
Derigs, D., Gassner, G. J., Walch, S. & Winters, A. R. (2018). Entropy Stable Finite Volume Approximations for Ideal Magnetohydrodynamics. Jahresbericht der Deutschen Mathematiker-Vereinigung (Teubner), 120(3), 153-219
Open this publication in new window or tab >>Entropy Stable Finite Volume Approximations for Ideal Magnetohydrodynamics
2018 (English)In: Jahresbericht der Deutschen Mathematiker-Vereinigung (Teubner), ISSN 0012-0456, E-ISSN 1869-7135, Vol. 120, no 3, p. 153-219Article in journal (Refereed) Published
Abstract [en]

This article serves as a summary outlining the mathematical entropy analysis of the ideal magnetohydrodynamic (MHD) equations. We select the ideal MHD equations as they are particularly useful for mathematically modeling a wide variety of magnetized fluids. In order to be self-contained we first motivate the physical properties of a magnetic fluid and how it should behave under the laws of thermodynamics. Next, we introduce a mathematical model built from hyperbolic partial differential equations (PDEs) that translate physical laws into mathematical equations. After an overview of the continuous analysis, we thoroughly describe the derivation of a numerical approximation of the ideal MHD system that remains consistent to the continuous thermodynamic principles. The derivation of the method and the theorems contained within serve as the bulk of the review article. We demonstrate that the derived numerical approximation retains the correct entropic properties of the continuous model and show its applicability to a variety of standard numerical test cases for MHD schemes. We close with our conclusions and a brief discussion on future work in the area of entropy consistent numerical methods and the modeling of plasmas.

Place, publisher, year, edition, pages
Springer Berlin/Heidelberg, 2018
Keywords
Computational physics, Entropy conservation, Entropy stability, Ideal MHD equa- tions, Finite volume methods
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-156852 (URN)10.1365/s13291-018-0178-9 (DOI)
Funder
German Research Foundation (DFG), SPP 1573EU, European Research Council, 714487EU, European Research Council, 679852
Available from: 2019-05-14 Created: 2019-05-14 Last updated: 2019-05-23Bibliographically approved
Derigs, D., Winters, A. R., Gassner, G. J., Walch, S. & Bohm, M. (2018). Ideal GLM-MHD: About the entropy consistent nine-wave magnetic field divergence diminishing ideal magnetohydrodynamics equations. Journal of Computational Physics, 364, 420-467
Open this publication in new window or tab >>Ideal GLM-MHD: About the entropy consistent nine-wave magnetic field divergence diminishing ideal magnetohydrodynamics equations
Show others...
2018 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 364, p. 420-467Article in journal (Refereed) Published
Abstract [en]

The paper presents two contributions in the context of the numerical simulation of magnetized fluid dynamics. First, we show how to extend the ideal magnetohydrodynamics (MHD) equations with an inbuilt magnetic field divergence cleaning mechanism in such a way that the resulting model is consistent with the second law of thermodynamics. As a byproduct of these derivations, we show that not all of the commonly used divergence cleaning extensions of the ideal MHD equations are thermodynamically consistent. Secondly, we present a numerical scheme obtained by constructing a specific finite volume discretization that is consistent with the discrete thermodynamic entropy. It includes a mechanism to control the discrete divergence error of the magnetic field by construction and is Galilean invariant. We implement the new high-order MHD solver in the adaptive mesh refinement code FLASH where we compare the divergence cleaning efficiency to the constrained transport solver available in FLASH (unsplit staggered mesh scheme).

Place, publisher, year, edition, pages
Elsevier, 2018
Keywords
magnetohydrodynamics, entropy stability, divergence-free magnetic field, divergence cleaning
National Category
Computational Mathematics Other Physics Topics
Identifiers
urn:nbn:se:liu:diva-156853 (URN)10.1016/j.jcp.2018.03.002 (DOI)000432481000020 ()2-s2.0-85045397430 (Scopus ID)
Funder
EU, European Research Council, 679852EU, European Research Council, 714487German Research Foundation (DFG), SPP 1573
Available from: 2019-05-14 Created: 2019-05-14 Last updated: 2019-05-23Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0002-5902-1522

Search in DiVA

Show all publications