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

Abstract

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 2D 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 AREPO-type approach, which rapidly rebuilds a new high quality mesh rearranging the element shapes and neighbors in order to guarantee a robust mesh evolution even for vortex flows and very long simulation times. The old and new Voronoi elements associated to the same generator are connected to construct closed space--time control volumes, whose bottom and top faces may be polygons with a different number of sides. We also incorporate degenerate space--time sliver elements, needed to fill the space--time holes that arise because of topology changes. The final ALE FV-DG scheme is obtained by a redesign of the fully discrete direct ALE schemes of Boscheri and Dumbser, extended here to moving Voronoi meshes and space--time sliver elements. Our new numerical scheme is based on the integration over arbitrary shaped closed space--time control volumes combined with a fully-discrete space--time conservation formulation of the governing PDE system. In this way the discrete solution is conservative and satisfies the GCL by construction. Numerical convergence studies as well as a large set of benchmarks 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 with topology changes lead to substantial improvements compared to direct ALE methods on conforming meshes.

References in corpus (26)

• E pur si muove: Galiliean-invariant cosmological hydrodynamical simulations on a moving mesh
• The PLUTO Code for Adaptive Mesh Computations in Astrophysical Fluid Dynamics
• Improving the convergence properties of the moving-mesh code AREPO
• A Posteriori Subcell Limiting of the Discontinuous Galerkin Finite Element Method for Hyperbolic Conservation Laws
• Second Order Accurate Schemes for Magnetohydrodynamics With Divergence-Free Reconstruction
• High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: viscous heat-conducting fluids and elastic solids
• Magnetic fields in cosmological simulations of disk galaxies
• Space-time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting
• Lagrangian ADER-WENO Finite Volume Schemes on Unstructured Tetrahedral Meshes for Conservative and Nonconservative Hyperbolic Systems in 3D
• Adaptive Mesh Refinement for Hyperbolic Systems based on Third-Order Compact WENO Reconstruction
• Central WENO schemes for hyperbolic conservation laws on fixed and moving unstructured meshes
• CWENO: uniformly accurate reconstructions for balance laws
• Arbitrary-Lagrangian-Eulerian discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes
• Arbitrary-Lagrangian-Eulerian One-Step WENO Finite Volume Schemes on Unstructured Triangular Meshes
• High order ADER schemes for a unified first order hyperbolic formulation of Newtonian continuum mechanics coupled with electro-dynamics
• High Order Lagrangian ADER-WENO Schemes on Unstructured Meshes - Application of Several Node Solvers to Hydrodynamics and Magnetohydrodynamics
• Space-time adaptive ADER-DG schemes for dissipative flows: compressible Navier-Stokes and resistive MHD equations
• Arbitrary order finite volume well-balanced schemes for the Euler equations with gravity
• Well balanced Arbitrary-Lagrangian-Eulerian finite volume schemes on moving nonconforming meshes for the Euler equations of gasdynamics with gravity
• Direct Arbitrary-Lagrangian-Eulerian finite volume schemes on moving nonconforming unstructured meshes
• On GLM curl cleaning for a first order reduction of the CCZ4 formulation of the Einstein field equations
• Arbitrary-Lagrangian-Eulerian ADER-WENO Finite Volume Schemes with Time-Accurate Local Time Stepping for Hyperbolic Conservation Laws
• A projection hybrid high order finite volume/finite element method for incompressible turbulent flows
• High Order Cell-Centered Lagrangian-Type Finite Volume Schemes with Time-Accurate Local Time Stepping on Unstructured Triangular Meshes
• On the Eigenvalues of the ADER-WENO Galerkin Predictor
• High order well-balanced finite volume methods for multi-dimensional systems of hyperbolic balance laws

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• An Arbitrary-Lagrangian-Eulerian hybrid finite volume/finite element method on moving unstructured meshes for the Navier-Stokes equations
• An Arbitrary High Order and Positivity Preserving Method for the Shallow Water Equations
• Efficient iterative arbitrary high order methods: an adaptive bridge between low and high order
• The eXtreme Mesh deformation approach (X-MESH) for the Stefan phase-change model
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• Space-time adaptive ADER-DG finite element method with LST-DG predictor and a posteriori sub-cell WENO finite-volume limiting for simulation of non-stationary compressible multicomponent reactive flows
• On improving the efficiency of ADER methods
• Well balanced finite volume schemes for shallow water equations on manifolds
• A Velocity-based Moving Mesh Virtual Element Method
• Space-time adaptive ADER-DG finite element method with LST-DG predictor and a posteriori sub-cell ADER-WENO finite-volume limiting for multidimensional detonation waves simulation
• High order well-balanced Arbitrary-Lagrangian-Eulerian ADER discontinuous Galerkin schemes on general polygonal moving meshes
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• High order ADER-DG method with local DG predictor for solutions of differential-algebraic systems of equations
• An Efficient Sliding Mesh Interface Method for High-Order Discontinuous Galerkin Schemes
• An efficient implicit scheme for the multimaterial Euler equations in Lagrangian coordinates
• Improving the local solution of the DG predictor of the ADER-DG method for solving systems of ordinary differential equations and its applicability to systems of differential-algebraic equations