An efficient high order direct ALE ADER finite volume scheme with a posteriori limiting for hydrodynamics and magnetohydrodynamics

Boscheri, Walter

doi:10.1002/fld.4342

“…A more recent and more sophisticated methodology to fulfill this condition exactly on the discrete level also in the context of high order ADER WENO finite volume schemes on unstructured simplex meshes can be found in [135]. A similar approach is adopted in [136,55,137].…”

Section: Ideal Mhd Equationsmentioning

confidence: 99%

High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes

Gaburro

¹

,

Boscheri

²

,

Chiocchetti

³

et al. 2020

Journal of Computational Physics

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We present a new family of very high order accurate direct Arbitrary-Lagrangian-Eulerian (ALE) Finite Volume (FV) and Discontinuous Galerkin (DG) schemes for the solution of nonlinear hyperbolic PDE systems on moving Voronoi meshes that are regenerated at each time step and which explicitly allow topology changes in time. The Voronoi tessellations are obtained from a set of generator points that move with the local fluid velocity. We employ an AREPOtype approach [1], which rapidly rebuilds a new high quality mesh exploiting the previous one, but rearranging the element shapes and neighbors in order to guarantee that the mesh evolution is robust even for vortex flows and for very long computational times. The old and new Voronoi elements associated to the same generator point are connected in space-time to construct closed space-time control volumes, whose bottom and top faces may be polygons with a different number of sides. We also need to incorporate some degenerate space-time sliver elements, which are needed in order to fill the space-time holes that arise because of the topology changes in the mesh between time t n and time t n+1 . The final ALE FV-DG scheme is obtained by a novel redesign of the high order accurate fully discrete direct ALE schemes of Boscheri and Dumbser [2,3], which have been extended here to general moving Voronoi meshes and space-time sliver elements. Our new numerical scheme is based on the integration over arbitrary shaped closed spacetime control volumes combined with a fully-discrete space-time conservation formulation of the governing hyperbolic PDE system. In this way the discrete solution is conservative and satisfies the geometric conservation law (GCL) by construction. Numerical convergence studies as well as a large set of benchmark problems for hydrodynamics and magnetohydrodynamics (MHD) demonstrate the accuracy and robustness of the proposed method. Our numerical results clearly show that the new combination of very high order schemes with regenerated meshes that allow topology changes in each time step lead to substantial improvements over the existing state of the art in direct ALE methods.Keywords: Arbitrary-Lagrangian-Eulerian (ALE) Finite Volume (FV) and Discontinuous Galerkin (DG) schemes, arbitrary high order in space and time, moving Voronoi tessellations with topology change, a posteriori sub-cell finite volume limiter, fully-discrete one-step ADER approach for hyperbolic PDE, compressible Euler and MHD equations directly integrated by means of a high order fully discrete one-step ADER method. To the best knowledge of the authors, this is the first time that arbitrary high order accurate direct ALE FV and DG schemes are developed with an embedded mesh generator that builds a new mesh with a different topology at each time step. State of the artLagrangian algorithms [4,5,6,7,8,9,10,11,12] are characterized by a moving computational mesh displaced with a velocity chosen as close as possible to the local fluid velocity. In the Lagrangian description of the fluid, the ...

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“…Here, (39) includes one additional scalar PDE for the evolution of the variable ψ, which is needed to transport divergence errors outside the computational domain with an artificial divergence cleaning speed c h , see [31,91]. A similar approach is adopted in [16,20,57].…”

Section: Ideal Mhd Equationsmentioning

confidence: 99%

A Posteriori Subcell Finite Volume Limiter for General $$P_NP_M$$ Schemes: Applications from Gasdynamics to Relativistic Magnetohydrodynamics

Gaburro

¹

,

Dumbser

²

2021

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In this work, we consider the general family of the so called ADER $$P_NP_M$$ P N P M schemes for the numerical solution of hyperbolic partial differential equations with arbitrary high order of accuracy in space and time. The family of one-step $$P_NP_M$$ P N P M schemes was introduced in Dumbser (J Comput Phys 227:8209–8253, 2008) and represents a unified framework for classical high order Finite Volume (FV) schemes ($$N=0$$ N = 0 ), the usual Discontinuous Galerkin (DG) methods ($$N=M$$ N = M ), as well as a new class of intermediate hybrid schemes for which a reconstruction operator of degree M is applied over piecewise polynomial data of degree N with $$M>N$$ M > N . In all cases with $$M \ge N > 0 $$ M ≥ N > 0 the $$P_NP_M$$ P N P M schemes are linear in the sense of Godunov (Math. USSR Sbornik 47:271–306, 1959), thus when considering phenomena characterized by discontinuities, spurious oscillations may appear and even destroy the simulation. Therefore, in this paper we present a new simple, robust and accurate a posteriori subcell finite volume limiting strategy that is valid for the entire class of $$P_NP_M$$ P N P M schemes. The subcell FV limiter is activated only where it is needed, i.e. in the neighborhood of shocks or other discontinuities, and is able to maintain the resolution of the underlying high order $$P_NP_M$$ P N P M schemes, due to the use of a rather fine subgrid of $$2N+1$$ 2 N + 1 subcells per space dimension. The paper contains a wide set of test cases for different hyperbolic PDE systems, solved on adaptive Cartesian meshes that show the capabilities of the proposed method both on smooth and discontinuous problems, as well as the broad range of its applicability. The tests range from compressible gasdynamics over classical MHD to relativistic magnetohydrodynamics.

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“…Here, (39) includes one additional scalar PDE for the evolution of the variable ψ, which is needed to transport divergence errors outside the computational domain with an artificial divergence cleaning speed c h , see [88,31]. A similar approach is adopted in [54,20,16]. A more recent and more sophisticated methodology to fulfill this condition exactly at the discrete level also in the context of high order ADER WENO finite volume schemes on unstructured simplex meshes can be found in [7].…”

Section: Ideal Mhd Equationsmentioning

confidence: 99%

A posteriori subcell finite volume limiter for general PNPM schemes: applications from gasdynamics to relativistic magnetohydrodynamics

Gaburro¹,

Dumbser²

2020

Preprint

0

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In this work, we consider the general family of the so called ADER P N P M schemes for the numerical solution of hyperbolic partial differential equations with arbitrary high order of accuracy in space and time.The family of one-step P N P M schemes was introduced in [39] and represents a unified framework for classical high order Finite Volume (FV) schemes (N = 0), the usual Discontinuous Galerkin (DG) methods (N = M ), as well as a new class of intermediate hybrid schemes for which a reconstruction operator of degree M is applied over piecewise polynomial data of degree N with M > N . In all cases with M ≥ N > 0 the P N P M schemes are linear in the sense of Godunov [61], thus when considering phenomena characterized by discontinuities, spurious oscillations may appear and even destroy the simulation. Therefore, in this paper we present a new simple, robust and accurate a posteriori subcell finite volume limiting strategy that is valid for the entire class of P N P M schemes. The subcell FV limiter is activated only where it is needed, i.e. in the neighborhood of shocks or other discontinuities, and is able to maintain the resolution of the underlying high order P N P M schemes, due to the use of a rather fine subgrid of 2N + 1 subcells per space dimension.The paper contains a wide set of test cases for different hyperbolic PDE systems, solved on adaptive Cartesian meshes (AMR) that show the capabilities of the proposed method both on smooth and discontinuous problems, as well as the broad range of its applicability. The tests range from compressible gasdynamics over classical MHD to relativistic magnetohydrodynamics.

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An efficient high order direct ALE ADER finite volume scheme with a posteriori limiting for hydrodynamics and magnetohydrodynamics

Cited by 10 publications

References 109 publications

High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes

High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes

A Posteriori Subcell Finite Volume Limiter for General $$P_NP_M$$ Schemes: Applications from Gasdynamics to Relativistic Magnetohydrodynamics

A posteriori subcell finite volume limiter for general PNPM schemes: applications from gasdynamics to relativistic magnetohydrodynamics

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