| 1. |
- Bohm, Marvin, et al.
(författare)
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An entropy stable nodal discontinuous Galerkin method for the resistive MHD equations. Part I : Theory and numerical verification
- 2018
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Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716.
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 2. |
- Bohm, Marvin, et al.
(författare)
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Multi-element SIAC Filter for Shock Capturing Applied to High-Order Discontinuous Galerkin Spectral Element Methods
- 2019
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Ingår i: Journal of Scientific Computing. - : Springer-Verlag New York. - 0885-7474 .- 1573-7691. ; 81:2, s. 820-844
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 3. |
- Derigs, Dominik, et al.
(författare)
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A novel averaging technique for discrete entropy-stable dissipation operators for ideal MHD
- 2017
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Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716. ; 330, s. 624-632
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Tidskriftsartikel (refereegranskat)abstract
- Entropy stable schemes can be constructed with a specific choice of the numerical flux function. First, an entropy conserving flux is constructed. Secondly, an entropy stable dissipation term is added to this flux to guarantee dissipation of the discrete entropy. Present works in the field of entropy stable numerical schemes are concerned with thorough derivations of entropy conservative fluxes for ideal MHD. However, as we show in this work, if the dissipation operator is not constructed in a very specific way, it cannot lead to a generally stable numerical scheme. The two main findings presented in this paper are that the entropy conserving flux of Ismail & Roe can easily break down for certain initial conditions commonly found in astrophysical simulations, and that special care must be taken in the derivation of a discrete dissipation matrix for an entropy stable numerical scheme to be robust. We present a convenient novel averaging procedure to evaluate the entropy Jacobians of the ideal MHD and the compressible Euler equations that yields a discretization with favorable robustness properties.
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| 4. |
- Derigs, Dominik, et al.
(författare)
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A novel high-order, entropy stable, 3D AMR MHD solver with guaranteed positive pressure
- 2016
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Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716. ; 317, s. 223-256
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Tidskriftsartikel (refereegranskat)abstract
- We describe a high-order numerical magnetohydrodynamics (MHD) solver built upon a novel non-linear entropy stable numerical flux function that supports eight travelling wave solutions. By construction the solver conserves mass, momentum, and energy and is entropy stable. The method is designed to treat the divergence-free constraint on the magnetic field in a similar fashion to a hyperbolic divergence cleaning technique. The solver described herein is especially well-suited for flows involving strong discontinuities. Furthermore, we present a new formulation to guarantee positivity of the pressure. We present the underlying theory and implementation of the new solver into the multi-physics, multi-scale adaptive mesh refinement (AMR) simulation code FLASH (http://flash.uchicago.edu). The accuracy, robustness and computational efficiency is demonstrated with a number of tests, including comparisons to available MHD implementations in FLASH.
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| 5. |
- Derigs, Dominik, et al.
(författare)
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Entropy Stable Finite Volume Approximations for Ideal Magnetohydrodynamics
- 2018
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Ingår i: Jahresbericht der Deutschen Mathematiker-Vereinigung (Teubner). - : Springer Berlin/Heidelberg. - 0012-0456 .- 1869-7135. ; 120:3, s. 153-219
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 6. |
- Derigs, Dominik, et al.
(författare)
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Ideal GLM-MHD : About the entropy consistent nine-wave magnetic field divergence diminishing ideal magnetohydrodynamics equations
- 2018
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Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716. ; 364, s. 420-467
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Tidskriftsartikel (refereegranskat)abstract
- 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).
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| 7. |
- Fernandez, David C. Del Rey, et al.
(författare)
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Entropy-stable p-nonconforming discretizations with the summation-by-parts property for the compressible Navier-Stokes equations
- 2020
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Ingår i: Computers & Fluids. - : PERGAMON-ELSEVIER SCIENCE LTD. - 0045-7930 .- 1879-0747. ; 210
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Tidskriftsartikel (refereegranskat)abstract
- The entropy-conservative/stable, curvilinear, nonconforming, p-refinement algorithm for hyperbolic conservation laws of Del Rey Fernandez et al. (2019) is extended from the compressible Euler equations to the compressible Navier-Stokes equations. A simple and flexible coupling procedure with planar interpolation operators between adjoining nonconforming elements is used. Curvilinear volume metric terms are numerically approximated via a minimization procedure and satisfy the discrete geometric conservation law conditions. Distinct curvilinear surface metrics are used on the adjoining interfaces to construct the interface coupling terms, thereby localizing the discrete geometric conservation law constraints to each individual element. The resulting scheme is entropy conservative/stable, element-wise conservative, and freestream preserving. Viscous interface dissipation operators that retain the entropy stability of the base scheme are developed. The accuracy and stability of the resulting numerical scheme are shown to be comparable to those of the original conforming scheme in Carpenter et al. (2014) and Parsani et al. (2016), i.e., this scheme achieves similar to p 1/2 convergence on geometrically high-order distorted element grids; this is demonstrated in the context of the viscous shock problem, the Taylor-Green vortex problem at a Reynolds number of Re = 1, 600, and a subsonic turbulent flow past a sphere at Re = 2, 000. (C) 2020 Published by Elsevier Ltd.
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| 8. |
- Friedrich, Lucas, et al.
(författare)
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An Entropy Stable h/p Non-Conforming Discontinuous Galerkin Method with the Summation-by-Parts Property
- 2018
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Ingår i: Journal of Scientific Computing. - : Springer. - 0885-7474 .- 1573-7691. ; 77:2, s. 689-725
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 9. |
- Friedrich, Lucas, et al.
(författare)
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Conservative and stable degree preserving SBP operators for non-conforming meshes
- 2018
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Ingår i: Journal of Scientific Computing. - : Springer-Verlag New York. - 0885-7474 .- 1573-7691. ; 75:2, s. 657-686
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 10. |
- Friedrich, Lucas, et al.
(författare)
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Entropy Stable Space-Time Discontinuous Galerkin Schemes with Summation-by-Parts Property for Hyperbolic Conservation Laws
- 2019
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Ingår i: Journal of Scientific Computing. - : Springer. - 0885-7474 .- 1573-7691. ; 80:1, s. 175-222
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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