Continuous finite element subgrid basis functions for Discontinuous Galerkin schemes on unstructured polygonal Voronoi meshes

We propose a new high order accurate nodal discontinuous Galerkin (DG) method for the solution of nonlinear hyperbolic systems of partial differential equations (PDE) on unstructured polygonal Voronoi meshes. Rather than using classical polynomials of degree N inside each element, in our new approach the discrete solution is represented by piecewise continuous polynomials of degree N within each Voronoi element, using a continuous finite element basis defined on a subgrid inside each polygon. We call the resulting subgrid basis an agglomerated finite element (AFE) basis for the DG method on general polygons, since it is obtained by the agglomeration of the finite element basis functions associated with the subgrid triangles. The basis functions on each sub-triangle are defined, as usual, on a universal reference element, hence allowing to compute universal mass, flux and stiffness matrices for the subgrid triangles once and for all in a pre-processing stage for the reference element only. Consequently, the construction of an efficient quadrature-free algorithm is possible, despite the unstructured nature of the computational grid. High order of accuracy in time is achieved thanks to the ADER approach, making use of an element-local space-time Galerkin finite element predictor. The novel schemes are carefully validated against a set of typical benchmark problems for the compressible Euler and Navier-Stokes equations. The numerical results have been checked with reference solutions available in literature and also systematically compared, in terms of computational efficiency and accuracy, with those obtained by the corresponding modal DG version of the scheme

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

Solving the system of radiation magnetohydrodynamics for solar physical simulations in 3d

In this study we present a finite-volumen scheme for solving the equations of radiation magnetohydrodynamics in two and three space dimensions. Among other applications this system is used to model the plasma in the solar convection zone and in the solar photosphere. It is a non--linear system of balance laws derived from the Euler equations of gas dynamics and the Maxwell equations; the energy transport through radiation is also included in the model. The starting point of our presentation is a standard explicit first and second order finite-volume scheme on both structured and unstructured grids. We first study the convergence of a finite-volume scheme applied to a scalar model problem for the full system of radiation magnetohydrodynamics. We then present modifications of the base scheme. These make it possible to approximate the system of magnetohydrodynamics with an arbitrary equation of state; they reduce errors due to a violation of the divergence constraint on the magnetic field, and they lead to an improved accuracy in the approximation of solution near an equilibrium state. These modifications significantly increase the robustness of the scheme and are essential for an accurate simulation of processes in the solar atmosphere ...thesi

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