ADER discontinuous Galerkin schemes for general-relativistic ideal magnetohydrodynamics

Fambri, Francesco; Dumbser, Michael; Köppel, Sven; Rezzolla, Luciano; Zanotti, Olindo

doi:10.1093/mnras/sty734

Cited by 53 publications

(84 citation statements)

References 123 publications

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“…A. A fixed Cartesian mesh made of N P quadrilaterals elements, which is not moved during the simulation, but which can be successively refined, with a general space-tree-type data structure that allows elementby-element refinement with a general refinement factor r ≥ 2, in order to increase the resolution in the areas of interest, as can be seen in Figure 2 (for the details on the refinement procedure we refer to [80,82]). To ease the description of the numerical method, we will associate to each quadrilateral element P n i , a set of indices that refer to its Cartesian coordinates, {j, k}, such that P n jk :…”

Section: Domain Discretizationmentioning

confidence: 99%

High Order ADER Schemes for Continuum Mechanics

et al. 2020

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In this paper we first review the development of high order ADER finite volume and ADER discontinuous Galerkin schemes on fixed and moving meshes, since their introduction in 1999 by Toro et al. We show the modern variant of ADER based on a space-time predictor-corrector formulation in the context of ADER discontinuous Galerkin schemes with a posteriori subcell finite volume limiter on fixed and moving grids, as well as on space-time adaptive Cartesian AMR meshes. We then present and discuss the unified symmetric hyperbolic and thermodynamically compatible (SHTC) formulation of continuum mechanics developed by Godunov, Peshkov and Romenski (GPR model), which allows to describe fluid and solid mechanics in one single and unified first order hyperbolic system. In order to deal with free surface and moving boundary problems, a simple diffuse interface approach is employed, which is compatible with Eulerian schemes on fixed grids as well as direct Arbitrary-Lagrangian-Eulerian methods on moving meshes. We show some examples of moving boundary problems in fluid and solid mechanics.

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Section: Domain Discretizationmentioning

confidence: 99%

High Order ADER Schemes for Continuum Mechanics

et al. 2020

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“…Furthermore, in order to increase the resolution in the areas of interest, the ADER-DG scheme described above has been implemented on space-time adaptive Cartesian meshes, with a cell-by-cell refinement approach; for all the details we refer to [44,38,141,47,46,13,140,113].…”

Section: Adaptive Mesh Refinement (Amr)mentioning

confidence: 99%

A simple diffuse interface approach for compressible flows around moving solids of arbitrary shape based on a reduced Baer–Nunziato model

et al. 2020

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In this paper we propose a new diffuse interface model for the numerical simulation of inviscid compressible flows around fixed and moving solid bodies of arbitrary shape. The solids are assumed to be moving rigid bodies, without any elastic properties. The mathematical model is a simplified case of the seven-equation Baer-Nunziato model of compressible multi-phase flows. The resulting governing PDE system is a nonlinear system of hyperbolic conservation laws with non-conservative products. The geometry of the solid bodies is simply specified via a scalar field that represents the volume fraction of the fluid present in each control volume. This allows the discretization of arbitrarily complex geometries on simple uniform or adaptive Cartesian meshes. Inside the solid bodies, the fluid volume fraction is zero, while it is unitary inside the fluid phase. Due to the diffuse interface nature of the model, the volume fraction function can assume any value between zero and one in mixed cells that are occupied by both, fluid and solid.We also prove that at the material interface, i.e. where the volume fraction jumps from unity to zero, the normal component of the fluid velocity assumes the value of the normal component of the solid velocity. This result can be directly derived from the governing equations, either via Riemann invariants or from the generalized Rankine Hugoniot conditions according to the theory of Dal Maso, Le Floch and Murat [89], which justifies the use of a path-conservative approach for treating the non-conservative products.The governing partial differential equations of our new model are solved on simple uniform Cartesian grids via a high order path-conservative ADER discontinuous Galerkin (DG) finite element method with a posteriori sub-cell finite volume (FV) limiter. Since the numerical method is of the shock capturing type, the fluid-solid boundary is never explicitly tracked by the numerical method, neither via interface reconstruction, nor via mesh motion.The effectiveness of the proposed approach is tested on a set of different numerical test problems, including 1D Riemann problems as well as supersonic flows over fixed and moving rigid bodies.Key words: diffuse interface model, compressible flows over fixed and moving solids, immersed boundary method for compressible flows, arbitrary high-order discontinuous Galerkin schemes, a posteriori sub-cell finite volume limiter (MOOD), path-conservative schemes for hyperbolic PDE with non-conservative products,

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“…Simulations are run on the domain Ω = [0, 1] × [0, 1] until t = 0.5 using a fourth order ADER-DG scheme [15,16,38] with polynomial approximation degree N = 3 and 100 × 4 spatial elements. A fine grid reference solution is computed with a third order ADER-WENO finite volume scheme [14,16,38] on 1000 cells.…”

Section: Numerical Examplesmentioning

confidence: 99%

A new continuum model for general relativistic viscous heat-conducting media

Romenski

Peshkov

Dumbser

et al. 2020

Phil. Trans. R. Soc. A.

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The lack of formulation of macroscopic equations for irreversible dynamics of viscous heat-conducting media compatible with the causality principle of Einstein’s special relativity and the Euler–Lagrange structure of general relativity is a long-lasting problem. In this paper, we propose a possible solution to this problem in the framework of SHTC equations. The approach does not rely on postulates of equilibrium irreversible thermodynamics but treats irreversible processes from the non-equilibrium point of view. Thus, each transfer process is characterized by a characteristic velocity of perturbation propagation in the non-equilibrium state, as well as by an intrinsic time/length scale of the dissipative dynamics. The resulting system of governing equations is formulated as a first-order system of hyperbolic equations with relaxation-type irreversible terms. Via a formal asymptotic analysis, we demonstrate that classical transport coefficients such as viscosity, heat conductivity, etc., are recovered in leading terms of our theory as effective transport coefficients. Some numerical examples are presented in order to demonstrate the viability of the approach. This article is part of the theme issue ‘Fundamental aspects of nonequilibrium thermodynamics’.

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ADER discontinuous Galerkin schemes for general-relativistic ideal magnetohydrodynamics

Cited by 53 publications

References 123 publications

High Order ADER Schemes for Continuum Mechanics

High Order ADER Schemes for Continuum Mechanics

A simple diffuse interface approach for compressible flows around moving solids of arbitrary shape based on a reduced Baer–Nunziato model

A new continuum model for general relativistic viscous heat-conducting media

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