An entropy stable nodal discontinuous Galerkin method for the resistive MHD equations

We present a discretely entropy stable discontinuous Galerkin (DG) method for the resistive magneto-hydrodynamic (MHD) equations on three-dimensional curvilinear un- structured 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 propor- tional 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. We focus on the resistive MHD equations. Subsequently, our first result is a proof that the resistive terms are symmetric and positive-definite when formulated in entropy space as gradients of the entropy variables. This enables us to show that the entropy inequality holds for the resistive MHD equations. The 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. The discrete analysis relies on the summation-by-parts (SBP) property, which is satisfied by the DG spectral element method (DGSEM) with Legendre-Gauss-Lobatto (LGL) nodes. With the help of a resulting split form approximation and by incorporating specific dicretiza- tions of the non-conservative terms, we obtain an overall entropy conservative DG scheme for the ideal MHD equations. We extend the scheme to an entropy stable approximation by adding appropriate dissipation terms. Further, we provide a detailed derivation and analysis of the entropy stable discretization on three-dimensional curvilinear meshes. Although the divergence-free constraint is included in the non-conservative terms, the resulting method has no particular treatment to control the magnetic field divergence errors, which pollute the solution quality. Hence, we also extend the standard resistive MHD equations and the according DG approximation with a divergence cleaning mech- anism that is based on a generalized Lagrange multiplier (GLM). Moreover, we equip the resulting scheme with certain shock capturing methods in order to regularize the approximation in oscillatory regions close to discontinuities. We provide numerical examples that verify the theoretical properties of the entropy stable method. Also, we demonstrate the increased robustness of the entropy stable method with a series of challenging numerical results, before we finally apply it to a real space physics model describing atmospheric plasma interactions

Discontinuous Galerkin methods: exploiting superconvergence for improved time-stepping

The discontinuous Galerkin (DG) methods are one of the most extensively researched classes of numerical methods for solving partial dfferential equations that display convective or diffusive qualities and have been popularly adopted by the scientific and engineering communities as a method capable of achieving arbitrary orders of accuracy in space. The choice of numerical flux function plays a pivotal role in the successful construction of DG methods and has an intrinsic effect on the superconvergence properties. As an inherent property of the spatial discretisation, superconvergence can only be retained in the solution through a sensitive pairing with a time integrator. The results of the literature and of this work suggest that an improved pairing between the spatial and temporal discretisations is both desirable and possible. We perform analysis of three different but related manifestations of superconvergence: the local, super-accurate points themselves; the subsequent global extraction via the Smoothness-Increasing Accuracy-Conserving (SIAC) filters; and the spectral properties that quantify, in terms of dispersion and dissipation errors, how accurately waves are convected. In order to explore the effect of the numerical flux function on superconvergence, we consider a generalisation of the “natural" upwind choice for a Method of Lines solution to the linear advection equation: the upwind-biased flux. We prove that the method is locally superconvergent at roots of a linear combination of the left- and right-Radau polynomials dependent on the value of a flux parameter and that the use of SIAC filters is still able to draw out the superconvergence information and create a globally smooth and superconvergent solution. In exploring the coupling of DG with a time integrator, we introduce a new scheme to a class of multi-stage multi-derivative methods, following recent incorporation of local DG technologies to recover superconvergence and achieve improved wave propagation properties

Discontinuous Galerkin Spectral Element Methods for Astrophysical Flows in Multi-physics Applications

In engineering applications, discontinuous Galerkin methods (DG) have been proven to be a powerful and flexible class of high order methods for problems in computational fluid dynamics. However, the potential benefits of DG for applications in astrophysical contexts is still relatively unexplored in its entirety. To this day, a decent number of studies surveying DG for astrophysical flows have been conducted. But the adoption of DG by the astrophysics community is just beginning to gain traction and integration of DG into established, multi-physics simulation frameworks for comprehensive astrophysical modeling is still lacking. It is our firm believe, that the full potential of novel approaches for numerically solving the fluid equations only shows under the pressure of real-world simulations with all aspects of multi-physics, challenging flow configurations, resolution and runtime constraints, and efficiency metrics on high-performance systems involved. Thus, we see the pressing need to propel DG from the well-trodden path of cataloguing test results under "optimal" laboratory conditions towards the harsh and unforgiving environment of large-scale astrophysics simulations. Consequently, the core of this work is the development and deployment of a robust DG scheme solving the ideal magneto-hydrodynamics equations with multiple species on three-dimensional Cartesian grids with adaptive mesh refinement. We chose to implement DG within the venerable simulation framework FLASH, with a specific focus on multi-physics problems in astrophysics. This entails modifications of the vanilla DG scheme to make it fit seamlessly within FLASH in such a way that all other physics modules can be naturally coupled without additional implementation overhead. A key ingredient is that our DG scheme uses mean value data organized into blocks - the central data structure in FLASH. Having the opportunity to work on mean values, allows us to rely on a rock-solid, monotone Finite Volume (FV) scheme as "backup" whenever the high order DG method fails in cases when the flow gets too harsh. Finding ways to combine the two schemes in a fail-safe manner without loosing primary conservation while still maintaining high order accuracy for smooth, well-resolved flows involves a series of careful considerations, which we document in this thesis. The result of our work is a novel shock capturing scheme - a hybrid between FV and DG - with smooth transitions between low and high order fluxes according to solution smoothness estimators. We present extensive validations and test cases, specifically its interaction with multi-physics modules in FLASH such as (self-)gravity and radiative transfer. We also investigate the benefits and pitfalls of integrating end-to-end entropy stability into our numerical scheme, with special focus on highly compressible turbulent flows and shocks. Our implementation of DG in FLASH allows us to conduct preliminary yet comprehensive astrophysics simulations proving that our new solver is ready for assessments and investigations by the astrophysics community

Multi-element SIAC Filter for Shock Capturing Applied to High-Order Discontinuous Galerkin Spectral Element Methods

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.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]</p

Multi-element SIAC Filter for Shock Capturing Applied to High-Order Discontinuous Galerkin Spectral Element Methods

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.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]</p

ECOS 2012

The 8-volume set contains the Proceedings of the 25th ECOS 2012 International Conference, Perugia, Italy, June 26th to June 29th, 2012. ECOS is an acronym for Efficiency, Cost, Optimization and Simulation (of energy conversion systems and processes), summarizing the topics covered in ECOS: Thermodynamics, Heat and Mass Transfer, Exergy and Second Law Analysis, Process Integration and Heat Exchanger Networks, Fluid Dynamics and Power Plant Components, Fuel Cells, Simulation of Energy Conversion Systems, Renewable Energies, Thermo-Economic Analysis and Optimisation, Combustion, Chemical Reactors, Carbon Capture and Sequestration, Building/Urban/Complex Energy Systems, Water Desalination and Use of Water Resources, Energy Systems- Environmental and Sustainability Issues, System Operation/ Control/Diagnosis and Prognosis, Industrial Ecology