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ADER discontinuous Galerkin schemes for general-relativistic ideal magnetohydrodynamics

Monthly notices of the Royal Astronomical Society, ISSN 1365-2966, 2018, Volume 477, Issue 4, pp. 4543 - 4564

Abstract We present a new class of high-order accurate numerical algorithms for solving the equations of general-relativistic ideal magnetohydrodynamics in...

Shock waves | Black hole physics -MHD- relativistic processes | Methods: numerical | ELEMENT-METHOD | RIEMANN SOLVER | ADAPTIVE MESH REFINEMENT | MHD | 1ST-ORDER HYPERBOLIC FORMULATION | COMPRESSIBLE NAVIER-STOKES | HIGH-ORDER | methods: numerical | relativistic processes | black hole physics | ASTRONOMY & ASTROPHYSICS | CONSERVATION-LAWS | UNSTRUCTURED MESHES | WENO LIMITERS | FINITE-VOLUME SCHEMES | shock waves

Shock waves | Black hole physics -MHD- relativistic processes | Methods: numerical | ELEMENT-METHOD | RIEMANN SOLVER | ADAPTIVE MESH REFINEMENT | MHD | 1ST-ORDER HYPERBOLIC FORMULATION | COMPRESSIBLE NAVIER-STOKES | HIGH-ORDER | methods: numerical | relativistic processes | black hole physics | ASTRONOMY & ASTROPHYSICS | CONSERVATION-LAWS | UNSTRUCTURED MESHES | WENO LIMITERS | FINITE-VOLUME SCHEMES | shock waves

Journal Article

Journal of computational physics, ISSN 0021-9991, 2008, Volume 227, Issue 18, pp. 8209 - 8253

.... The time evolution of these data and the flux computation, however, are then done with a different set of piecewise polynomials of degree M...

Unstructured meshes | Discontinuous Galerkin | Local continuous space–time Galerkin method | Ideal and relativistic MHD equations | Hyperbolic PDE | M-exact [formula omitted] least squares reconstruction | WENO | One-step time discretization | Finite volume | Euler equations | Nonlinear elasticity | least squares reconstruction | M-exact P | Local continuous space-time Galerkin method | ELEMENT-METHOD | hyperbolic PDE | COMPRESSIBLE MEDIUM | PHYSICS, MATHEMATICAL | nonlinear elasticity | discontinuous Galerkin | CONSERVATION-LAWS | PNPM | GENERALIZED RIEMANN PROBLEM | TANG VORTEX SYSTEM | EFFICIENT IMPLEMENTATION | ASYMPTOTIC-EXPANSION | HIGH-ORDER | finite volume | local continuous space-time Galerkin method | ideal and relativistic MHD equations | M-exact | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | ADER SCHEMES | DYNAMIC GRID MOTION | one-step time discretization | unstructured meshes

Unstructured meshes | Discontinuous Galerkin | Local continuous space–time Galerkin method | Ideal and relativistic MHD equations | Hyperbolic PDE | M-exact [formula omitted] least squares reconstruction | WENO | One-step time discretization | Finite volume | Euler equations | Nonlinear elasticity | least squares reconstruction | M-exact P | Local continuous space-time Galerkin method | ELEMENT-METHOD | hyperbolic PDE | COMPRESSIBLE MEDIUM | PHYSICS, MATHEMATICAL | nonlinear elasticity | discontinuous Galerkin | CONSERVATION-LAWS | PNPM | GENERALIZED RIEMANN PROBLEM | TANG VORTEX SYSTEM | EFFICIENT IMPLEMENTATION | ASYMPTOTIC-EXPANSION | HIGH-ORDER | finite volume | local continuous space-time Galerkin method | ideal and relativistic MHD equations | M-exact | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | ADER SCHEMES | DYNAMIC GRID MOTION | one-step time discretization | unstructured meshes

Journal Article

Computer methods in applied mechanics and engineering, ISSN 0045-7825, 2014, Volume 280, pp. 57 - 83

In this article a new high order accurate cell-centered Arbitrary-Lagrangian–Eulerian (ALE) Godunov-type finite volume method with time-accurate local time...

Hyperbolic conservation laws | Euler equations of compressible gas dynamics | Arbitrary-Lagrangian–Eulerian (ALE) Godunov-type finite volume methods | Time-accurate local time stepping (LTS) | High order Lagrangian ADER–WENO schemes | Magnetohydrodynamics equations (MHD) | High order Lagrangian ADER-WENO schemes | Arbitrary-Lagrangian-Eulerian (ALE) Godunov-type finite volume methods | TETRAHEDRAL MESHES | ADAPTIVE MESH REFINEMENT | EQUATIONS | DISCONTINUOUS GALERKIN METHOD | HIGH-ORDER | GAS-DYNAMICS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | COMPRESSIBLE FLOW | BALANCE LAWS | SYSTEMS | UNSTRUCTURED MESHES | Environmental law | Reconstruction | Formulations | Algorithms | Gas dynamics | Mathematical analysis | Conservation | Magnetohydrodynamic equations | Finite volume method | Mathematics - Numerical Analysis

Hyperbolic conservation laws | Euler equations of compressible gas dynamics | Arbitrary-Lagrangian–Eulerian (ALE) Godunov-type finite volume methods | Time-accurate local time stepping (LTS) | High order Lagrangian ADER–WENO schemes | Magnetohydrodynamics equations (MHD) | High order Lagrangian ADER-WENO schemes | Arbitrary-Lagrangian-Eulerian (ALE) Godunov-type finite volume methods | TETRAHEDRAL MESHES | ADAPTIVE MESH REFINEMENT | EQUATIONS | DISCONTINUOUS GALERKIN METHOD | HIGH-ORDER | GAS-DYNAMICS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | COMPRESSIBLE FLOW | BALANCE LAWS | SYSTEMS | UNSTRUCTURED MESHES | Environmental law | Reconstruction | Formulations | Algorithms | Gas dynamics | Mathematical analysis | Conservation | Magnetohydrodynamic equations | Finite volume method | Mathematics - Numerical Analysis

Journal Article

Axioms, ISSN 2075-1680, 2018, Volume 7, Issue 3, p. 63

In this paper we discuss a new and very efficient implementation of high order accurate arbitrary high order schemes using derivatives discontinuous Galerkin...

High performance computing | Hyperbolic partial differential equations | High order discontinuous Galerkin finite element schemes | Vectorization and parallelization | Shock waves and discontinuities | Magnetohydrodynamics | Compressibility | Propagation | Partial differential equations | Relativity | Elasticity | Finite element method | Conservation laws | Einstein equations | Mathematical analysis | Runge-Kutta method | Theory of relativity | Continuum mechanics | Mathematical models | Nonlinear systems | Fluid mechanics | Sound waves | Computational fluid dynamics | Numerical methods | Fluid | Supercomputers | Euler-Lagrange equation | Shock waves | Numerical analysis | Relativism | Galerkin method | Methods | hyperbolic partial differential equations | vectorization and parallelization | high performance computing | shock waves and discontinuities | high order discontinuous Galerkin finite element schemes

High performance computing | Hyperbolic partial differential equations | High order discontinuous Galerkin finite element schemes | Vectorization and parallelization | Shock waves and discontinuities | Magnetohydrodynamics | Compressibility | Propagation | Partial differential equations | Relativity | Elasticity | Finite element method | Conservation laws | Einstein equations | Mathematical analysis | Runge-Kutta method | Theory of relativity | Continuum mechanics | Mathematical models | Nonlinear systems | Fluid mechanics | Sound waves | Computational fluid dynamics | Numerical methods | Fluid | Supercomputers | Euler-Lagrange equation | Shock waves | Numerical analysis | Relativism | Galerkin method | Methods | hyperbolic partial differential equations | vectorization and parallelization | high performance computing | shock waves and discontinuities | high order discontinuous Galerkin finite element schemes

Journal Article

Continuum mechanics and thermodynamics, ISSN 1432-0959, 2019, Volume 31, Issue 5, pp. 1517 - 1541

This paper is an attempt to introduce methods and concepts of the Riemann–Cartan geometry largely used in such physical theories as general relativity, gauge...

Turbulence | Classical and Continuum Physics | Riemann–Cartan geometry | Engineering Thermodynamics, Heat and Mass Transfer | Hyperbolic equations | Theoretical and Applied Mechanics | Structural Materials | Torsion | Physics | DISLOCATIONS | STRETCH | 1ST-ORDER HYPERBOLIC FORMULATION | MODEL | Riemann-Cartan geometry | RELATIVISTIC THERMODYNAMICS | MECHANICS | THERMODYNAMICS | ORDER ADER SCHEMES | EINSTEIN | SYSTEMS | DERIVATIVES | FINITE-VOLUME SCHEMES | Fluid dynamics | Energy trading | Analysis | Viscosity | Solid mechanics | Electrodynamics | Degrees of freedom | Deformation | Turbulent flow | Computational fluid dynamics | Relativity | Fluid flow | Hydrodynamics | Distortion | Entropy | Dispersion | Tensors | Gauge theory | Mathematical analysis | Energy conservation | Energy dissipation | Continuum mechanics | Curvature | Rotating matter

Turbulence | Classical and Continuum Physics | Riemann–Cartan geometry | Engineering Thermodynamics, Heat and Mass Transfer | Hyperbolic equations | Theoretical and Applied Mechanics | Structural Materials | Torsion | Physics | DISLOCATIONS | STRETCH | 1ST-ORDER HYPERBOLIC FORMULATION | MODEL | Riemann-Cartan geometry | RELATIVISTIC THERMODYNAMICS | MECHANICS | THERMODYNAMICS | ORDER ADER SCHEMES | EINSTEIN | SYSTEMS | DERIVATIVES | FINITE-VOLUME SCHEMES | Fluid dynamics | Energy trading | Analysis | Viscosity | Solid mechanics | Electrodynamics | Degrees of freedom | Deformation | Turbulent flow | Computational fluid dynamics | Relativity | Fluid flow | Hydrodynamics | Distortion | Entropy | Dispersion | Tensors | Gauge theory | Mathematical analysis | Energy conservation | Energy dissipation | Continuum mechanics | Curvature | Rotating matter

Journal Article

SIAM journal on scientific computing, ISSN 1095-7197, 2017, Volume 39, Issue 6, pp. A2564 - A2591

We present a novel family of arbitrary high order accurate central Weighted ENO (CWENO) finite volume schemes for the solution of nonlinear systems of...

Arbitrary-Lagrangian-Eulerian finite volume schemes | High order in space | Fully discrete one-step ADER approach | Moving unstructured meshes | Central Weighted ENO reconstruction | Finite volume schemes on fixed | ALE | CWENO | Time | Large scale parallel high-performance computing computations | Hyperbolic conservation laws in multiple space dimensions | TRIANGULAR MESHES | MATHEMATICS, APPLIED | large scale parallel high-performance computing computations | fully discrete one-step ADER approach | NONUNIFORM MESHES | TETRAHEDRAL MESHES | ESSENTIALLY NONOSCILLATORY SCHEMES | high order in space and time | hyperbolic conservation laws in multiple space dimensions | HIGH-ORDER WENO | central Weighted ENO reconstruction | finite volume schemes on fixed and moving unstructured meshes | arbitrary-Lagrangian-Eulerian finite volume schemes | CONTINUUM-MECHANICS | ADER SCHEMES | BALANCE LAWS | CENTRAL WENO SCHEME | FINITE-VOLUME SCHEMES

Arbitrary-Lagrangian-Eulerian finite volume schemes | High order in space | Fully discrete one-step ADER approach | Moving unstructured meshes | Central Weighted ENO reconstruction | Finite volume schemes on fixed | ALE | CWENO | Time | Large scale parallel high-performance computing computations | Hyperbolic conservation laws in multiple space dimensions | TRIANGULAR MESHES | MATHEMATICS, APPLIED | large scale parallel high-performance computing computations | fully discrete one-step ADER approach | NONUNIFORM MESHES | TETRAHEDRAL MESHES | ESSENTIALLY NONOSCILLATORY SCHEMES | high order in space and time | hyperbolic conservation laws in multiple space dimensions | HIGH-ORDER WENO | central Weighted ENO reconstruction | finite volume schemes on fixed and moving unstructured meshes | arbitrary-Lagrangian-Eulerian finite volume schemes | CONTINUUM-MECHANICS | ADER SCHEMES | BALANCE LAWS | CENTRAL WENO SCHEME | FINITE-VOLUME SCHEMES

Journal Article

International journal for numerical methods in fluids, ISSN 0271-2091, 2019, Volume 89, Issue 1-2, pp. 16 - 42

Summary In this paper, we present a novel pressure‐based semi‐implicit finite volume solver for the equations of compressible ideal, viscous, and resistive...

ideal magnetohydrodynamics | viscous and resistive MHD | all Mach number flow solver | divergence‐free | finite volume schemes | semi‐implicit | compressible low Mach number flows | general equation of state | pressure‐based method | pressure-based method | semi-implicit | divergence-free | HLLC RIEMANN SOLVER | TANG VORTEX SYSTEM | THERMODYNAMIC PROPERTIES | PHYSICS, FLUIDS & PLASMAS | 1ST-ORDER HYPERBOLIC FORMULATION | GODUNOV-TYPE SCHEMES | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | NAVIER-STOKES EQUATIONS | ORDER ADER SCHEMES | DISCONTINUOUS GALERKIN METHODS | CONSERVATION-LAWS | UNSTRUCTURED MESHES | Fluid dynamics | Viscosity | Magnetohydrodynamics | Divergence | Compressibility | Methodology | Fluid flow | Finite volume method | Energy | Mathematical analysis | Solvers | Evolution | Mach number | Computational fluid dynamics | Momentum equation | Curl (vectors) | Dimensions | Momentum | Pressure | Velocity | Equations | Internal energy | Incompressible flow | Simulation | Energy equation | Volume | Newton methods | Computer applications | Scaling | Magnetic fields | Computing time | Riemann solver | Linear functions

ideal magnetohydrodynamics | viscous and resistive MHD | all Mach number flow solver | divergence‐free | finite volume schemes | semi‐implicit | compressible low Mach number flows | general equation of state | pressure‐based method | pressure-based method | semi-implicit | divergence-free | HLLC RIEMANN SOLVER | TANG VORTEX SYSTEM | THERMODYNAMIC PROPERTIES | PHYSICS, FLUIDS & PLASMAS | 1ST-ORDER HYPERBOLIC FORMULATION | GODUNOV-TYPE SCHEMES | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | NAVIER-STOKES EQUATIONS | ORDER ADER SCHEMES | DISCONTINUOUS GALERKIN METHODS | CONSERVATION-LAWS | UNSTRUCTURED MESHES | Fluid dynamics | Viscosity | Magnetohydrodynamics | Divergence | Compressibility | Methodology | Fluid flow | Finite volume method | Energy | Mathematical analysis | Solvers | Evolution | Mach number | Computational fluid dynamics | Momentum equation | Curl (vectors) | Dimensions | Momentum | Pressure | Velocity | Equations | Internal energy | Incompressible flow | Simulation | Energy equation | Volume | Newton methods | Computer applications | Scaling | Magnetic fields | Computing time | Riemann solver | Linear functions

Journal Article

Journal of scientific computing, ISSN 1573-7691, 2010, Volume 48, Issue 1-3, pp. 70 - 88

We propose a simple extension of the well-known Riemann solver of Osher and Solomon (Math. Comput. 38:339–374, 1982) to a certain class of hyperbolic systems...

Shallow water equations | Well-balanced schemes | Discontinuous Galerkin method | Computational Mathematics and Numerical Analysis | Path-conservative schemes | Theoretical, Mathematical and Computational Physics | Baer–Nunziato model | Multi-phase flows | Mathematics | Algorithms | Osher Riemann solver | Appl.Mathematics/Computational Methods of Engineering | P N P M schemes | Pitman & Le model | Finite volume schemes | schemes | Baer-Nunziato model | MATHEMATICS, APPLIED | COMPRESSIBLE 2-PHASE FLOW | PNPM schemes | DIFFERENCE-SCHEMES | SHALLOW-WATER EQUATIONS | GODUNOV METHOD | SOURCE TERMS | ORDER | PRODUCTS | CONSERVATION-LAWS | UNSTRUCTURED MESHES | FINITE-VOLUME SCHEMES | Computation | Topography | Entropy | Gaussian | Mathematical models | Formalism | Hyperbolic systems | Riemann solver

Shallow water equations | Well-balanced schemes | Discontinuous Galerkin method | Computational Mathematics and Numerical Analysis | Path-conservative schemes | Theoretical, Mathematical and Computational Physics | Baer–Nunziato model | Multi-phase flows | Mathematics | Algorithms | Osher Riemann solver | Appl.Mathematics/Computational Methods of Engineering | P N P M schemes | Pitman & Le model | Finite volume schemes | schemes | Baer-Nunziato model | MATHEMATICS, APPLIED | COMPRESSIBLE 2-PHASE FLOW | PNPM schemes | DIFFERENCE-SCHEMES | SHALLOW-WATER EQUATIONS | GODUNOV METHOD | SOURCE TERMS | ORDER | PRODUCTS | CONSERVATION-LAWS | UNSTRUCTURED MESHES | FINITE-VOLUME SCHEMES | Computation | Topography | Entropy | Gaussian | Mathematical models | Formalism | Hyperbolic systems | Riemann solver

Journal Article

Communications in computational physics, ISSN 1991-7120, 2015, Volume 16, Issue 3, pp. 718 - 763

In this paper, we investigate the coupling of the Multi-dimensional Optimal Order Detection (MOOD) method and the Arbitrary high order DERivatives (ADER)...

Relativistic MHD equations | Hyperbolic PDE | WENO | Euler equations | ADER | Finite Volume | Polynomial reconstruction | Unstructured meshes | Conservation law | MOOD | MHD equations | High-order | One-step time discretization | Local continuous space-time Galerkin method | ELEMENT-METHOD | MESHES | high-order | ESSENTIALLY NONOSCILLATORY SCHEMES | hyperbolic PDE | COMPRESSIBLE MEDIUM | PHYSICS, MATHEMATICAL | DIFFERENCE WENO SCHEMES | polynomial reconstruction | TANG VORTEX SYSTEM | EFFICIENT IMPLEMENTATION | DISCONTINUOUS GALERKIN METHOD | local continuous space-time Galerkin method | RIEMANN PROBLEM | conservation law | one-step time discretization | POSITIVITY | relativistic MHD equations | unstructured meshes

Relativistic MHD equations | Hyperbolic PDE | WENO | Euler equations | ADER | Finite Volume | Polynomial reconstruction | Unstructured meshes | Conservation law | MOOD | MHD equations | High-order | One-step time discretization | Local continuous space-time Galerkin method | ELEMENT-METHOD | MESHES | high-order | ESSENTIALLY NONOSCILLATORY SCHEMES | hyperbolic PDE | COMPRESSIBLE MEDIUM | PHYSICS, MATHEMATICAL | DIFFERENCE WENO SCHEMES | polynomial reconstruction | TANG VORTEX SYSTEM | EFFICIENT IMPLEMENTATION | DISCONTINUOUS GALERKIN METHOD | local continuous space-time Galerkin method | RIEMANN PROBLEM | conservation law | one-step time discretization | POSITIVITY | relativistic MHD equations | unstructured meshes

Journal Article

Computational astrophysics and cosmology, ISSN 2197-7909, 2016, Volume 3, Issue 1, pp. 1 - 32

We present a new version of conservative ADER-WENO finite volume schemes, in which both the high order spatial reconstruction as well as the time evolution of...

Computational Mathematics and Numerical Analysis | high order WENO reconstruction in primitive variables | ADER-WENO finite volume schemes | hyperbolic conservation laws | AMR | Numeric Computing | ADER discontinuous Galerkin schemes | relativistic hydrodynamics and magnetohydrodynamics | Physics | Astronomy, Astrophysics and Cosmology | Baer-Nunziato model | Reconstruction | Magnetohydrodynamics | Compressibility | Two phase flow | Computational fluid dynamics | Gas dynamics | Fluid flow | Hydrodynamics | Fluxes | Euler-Lagrange equation | Conversion | Relativistic effects | Finite element method | Accuracy | Spacetime | Relativism | Flow control | Evolution | Polynomials | Mathematical models | Hyperbolic systems | Research

Computational Mathematics and Numerical Analysis | high order WENO reconstruction in primitive variables | ADER-WENO finite volume schemes | hyperbolic conservation laws | AMR | Numeric Computing | ADER discontinuous Galerkin schemes | relativistic hydrodynamics and magnetohydrodynamics | Physics | Astronomy, Astrophysics and Cosmology | Baer-Nunziato model | Reconstruction | Magnetohydrodynamics | Compressibility | Two phase flow | Computational fluid dynamics | Gas dynamics | Fluid flow | Hydrodynamics | Fluxes | Euler-Lagrange equation | Conversion | Relativistic effects | Finite element method | Accuracy | Spacetime | Relativism | Flow control | Evolution | Polynomials | Mathematical models | Hyperbolic systems | Research

Journal Article

Journal of computational physics, ISSN 0021-9991, 2017, Volume 341, pp. 341 - 376

We propose a new arbitrary high order accurate semi-implicit space–time discontinuous Galerkin (DG) method for the solution of the two and three dimensional...

All Mach number flows | Pressure-based semi-implicit space–time discontinuous Galerkin scheme | Compressible Navier–Stokes equations | Staggered unstructured meshes | High order of accuracy in space and time | DIFFERENCE-SCHEMES | PHYSICS, MATHEMATICAL | Compressible Navier Stokes equations | SHALLOW-WATER EQUATIONS | GODUNOV-TYPE SCHEMES | FREE-SURFACE | NUMERICAL-SOLUTION | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | Pressure-based semi-implicit space time discontinuous Galerkin scheme | DYNAMIC GRID MOTION | VISCOUS INCOMPRESSIBLE-FLOW | FINITE-ELEMENT-METHOD | HYPERBOLIC CONSERVATION-LAWS | DRIVEN CAVITY FLOW | Fluid dynamics | Energy conservation | Statistics | Methods | Force and energy | Mathematics - Numerical Analysis

All Mach number flows | Pressure-based semi-implicit space–time discontinuous Galerkin scheme | Compressible Navier–Stokes equations | Staggered unstructured meshes | High order of accuracy in space and time | DIFFERENCE-SCHEMES | PHYSICS, MATHEMATICAL | Compressible Navier Stokes equations | SHALLOW-WATER EQUATIONS | GODUNOV-TYPE SCHEMES | FREE-SURFACE | NUMERICAL-SOLUTION | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | Pressure-based semi-implicit space time discontinuous Galerkin scheme | DYNAMIC GRID MOTION | VISCOUS INCOMPRESSIBLE-FLOW | FINITE-ELEMENT-METHOD | HYPERBOLIC CONSERVATION-LAWS | DRIVEN CAVITY FLOW | Fluid dynamics | Energy conservation | Statistics | Methods | Force and energy | Mathematics - Numerical Analysis

Journal Article