Arbitrary-Lagrangian–Eulerian ADER–WENO finite volume schemes with time-accurate local time stepping for hyperbolic conservation laws

Abstract In this article a new high order accurate cell-centered Arbitrary-Lagrangian–Eulerian (ALE) Godunov-type finite volume method with time-accurate local time stepping (LTS) is presented. The method is by construction locally and globally conservative. The scheme is based on a one-step predictor–corrector methodology in space–time and uses three main building blocks: First, a high order piecewise polynomial WENO reconstruction, to obtain a high order data representation in space from the known cell averages of the underlying finite volume scheme. Second, a high order space–time Galerkin predictor step based on a weak formulation of the governing PDE on moving control volumes. Third, a high order one-step finite volume scheme, based directly on the integral formulation of the conservation law in space–time. The algorithm being entirely based on space–time control volumes naturally allows for hanging nodes also in time, hence in this framework the implementation of a consistent and conservative time-accurate LTS becomes very natural and simple. The method is validated on some classical shock tube problems for the Euler equations of compressible gas dynamics and the magnetohydrodynamics equations (MHD). The performance of the new scheme is compared with a classical high order ALE finite volume scheme based on global time stepping. To the knowledge of the author, this is the first high order accurate Lagrangian finite volume method ever presented together with a conservative and time-accurate local time stepping feature.

[1]  Michael Dumbser,et al.  Arbitrary-Lagrangian-Eulerian One-Step WENO Finite Volume Schemes on Unstructured Triangular Meshes , 2013, 1302.3076.

[2]  Michael Dumbser,et al.  Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems , 2007, J. Comput. Phys..

[3]  Bruno Després,et al.  A cell-centered Lagrangian hydrodynamics scheme on general unstructured meshes in arbitrary dimension , 2009, J. Comput. Phys..

[4]  Michael Dumbser,et al.  Very high order PNPM schemes on unstructured meshes for the resistive relativistic MHD equations , 2009, J. Comput. Phys..

[5]  Pierre-Henri Maire,et al.  A high-order cell-centered Lagrangian scheme for compressible fluid flows in two-dimensional cylindrical geometry , 2009, J. Comput. Phys..

[6]  Pavel Váchal,et al.  Synchronized flux corrected remapping for ALE methods , 2011 .

[7]  Raphaël Loubère,et al.  A second-order compatible staggered Lagrangian hydrodynamics scheme using a cell-centered multidimensional approximate Riemann solver , 2010, ICCS.

[8]  Michael Dumbser,et al.  On Arbitrary-Lagrangian-Eulerian One-Step WENO Schemes for Stiff Hyperbolic Balance Laws , 2012, 1207.6407.

[9]  Guglielmo Scovazzi,et al.  A geometrically-conservative, synchronized, flux-corrected remap for arbitrary Lagrangian-Eulerian computations with nodal finite elements , 2011, J. Comput. Phys..

[10]  Michael Dumbser,et al.  ADER Schemes for Nonlinear Systems of Stiff Advection–Diffusion–Reaction Equations , 2011, J. Sci. Comput..

[11]  Jérôme Breil,et al.  Hybrid remap for multi-material ALE , 2011 .

[12]  Manuel Torrilhon,et al.  Non-uniform convergence of finite volume schemes for Riemann problems of ideal magnetohydrodynamics , 2003 .

[13]  Michael Dumbser,et al.  Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems , 2007, J. Comput. Phys..

[14]  Boleslaw K. Szymanski,et al.  Adaptive Local Refinement with Octree Load Balancing for the Parallel Solution of Three-Dimensional Conservation Laws , 1997, J. Parallel Distributed Comput..

[15]  Claus-Dieter Munz,et al.  A Discontinuous Galerkin Scheme based on a Space-Time Expansion II. Viscous Flow Equations in Multi Dimensions , 2008, J. Sci. Comput..

[16]  Dinshaw S. Balsara,et al.  Notes on the Eigensystem of Magnetohydrodynamics , 1996, SIAM J. Appl. Math..

[17]  Mikhail Shashkov,et al.  A finite volume cell‐centered Lagrangian hydrodynamics approach for solids in general unstructured grids , 2013 .

[18]  Claus-Dieter Munz,et al.  On Godunov-type schemes for Lagrangian gas dynamics , 1994 .

[19]  M. Dumbser,et al.  Heterogeneous Domain Decomposition for Computational Aeroacoustics , 2006 .

[20]  M. Dumbser,et al.  High order space–time adaptive ADER-WENO finite volume schemes for non-conservative hyperbolic systems , 2013, 1304.5408.

[21]  Eleuterio F. Toro,et al.  Space–time adaptive numerical methods for geophysical applications , 2009, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[22]  Pierre-Henri Maire,et al.  A unified sub‐cell force‐based discretization for cell‐centered Lagrangian hydrodynamics on polygonal grids , 2011 .

[23]  Michael Dumbser,et al.  Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier–Stokes equations , 2010 .

[24]  Eleuterio F. Toro,et al.  ADER: Arbitrary High Order Godunov Approach , 2002, J. Sci. Comput..

[25]  Lilia Krivodonova,et al.  An efficient local time-stepping scheme for solution of nonlinear conservation laws , 2010, J. Comput. Phys..

[26]  Marcus J. Grote,et al.  High-order explicit local time-stepping methods for damped wave equations , 2011, J. Comput. Appl. Math..

[27]  Chi-Wang Shu,et al.  High order conservative Lagrangian schemes with Lax-Wendroff type time discretization for the compressible Euler equations , 2009, J. Comput. Phys..

[28]  Bruno Després,et al.  Lagrangian Gas Dynamics in Two Dimensions and Lagrangian systems , 2005 .

[29]  Antonio Baeza,et al.  Adaptive mesh refinement techniques for high‐order shock capturing schemes for multi‐dimensional hydrodynamic simulations , 2006 .

[30]  Michael Dumbser,et al.  A direct Arbitrary-Lagrangian-Eulerian ADER-WENO finite volume scheme on unstructured tetrahedral meshes for conservative and non-conservative hyperbolic systems in 3D , 2014, J. Comput. Phys..

[31]  C. Munz,et al.  Hyperbolic divergence cleaning for the MHD equations , 2002 .

[32]  Pavel B. Bochev,et al.  Fast optimization-based conservative remap of scalar fields through aggregate mass transfer , 2013, J. Comput. Phys..

[33]  S. Osher,et al.  Uniformly high order accurate essentially non-oscillatory schemes, 111 , 1987 .

[34]  Chi-Wang Shu,et al.  A cell-centered Lagrangian scheme with the preservation of symmetry and conservation properties for compressible fluid flows in two-dimensional cylindrical geometry , 2010, J. Comput. Phys..

[35]  Raimund Bürger,et al.  Spectral WENO schemes with Adaptive Mesh Refinement for models of polydisperse sedimentation , 2013 .

[36]  Marcus J. Grote,et al.  Explicit local time-stepping methods for Maxwell's equations , 2010, J. Comput. Appl. Math..

[37]  Michael Dumbser,et al.  Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws , 2008, J. Comput. Phys..

[38]  Michael Dumbser,et al.  A Simple Extension of the Osher Riemann Solver to Non-conservative Hyperbolic Systems , 2011, J. Sci. Comput..

[39]  Eleuterio F. Toro,et al.  ADER schemes for three-dimensional non-linear hyperbolic systems , 2005 .

[40]  Raphaël Loubère,et al.  ReALE: A Reconnection Arbitrary-Lagrangian―Eulerian method in cylindrical geometry , 2011 .

[41]  Michael Dumbser,et al.  On Universal Osher-Type Schemes for General Nonlinear Hyperbolic Conservation Laws , 2011 .

[42]  Rémi Abgrall,et al.  A Cell-Centered Lagrangian Scheme for Two-Dimensional Compressible Flow Problems , 2007, SIAM J. Sci. Comput..

[43]  M. Dumbser,et al.  High-Order Unstructured Lagrangian One-Step WENO Finite Volume Schemes for Non-Conservative Hyperbolic Systems: Applications to Compressible Multi-Phase Flows , 2013, 1304.4816.

[44]  Chi-Wang Shu,et al.  A high order ENO conservative Lagrangian type scheme for the compressible Euler equations , 2007, J. Comput. Phys..

[45]  E. Toro,et al.  An arbitrary high-order Discontinuous Galerkin method for elastic waves on unstructured meshes - V. Local time stepping and p-adaptivity , 2007 .

[46]  Michael Dumbser,et al.  ADER-WENO finite volume schemes with space-time adaptive mesh refinement , 2012, J. Comput. Phys..

[47]  Guglielmo Scovazzi,et al.  Lagrangian shock hydrodynamics on tetrahedral meshes: A stable and accurate variational multiscale approach , 2012, J. Comput. Phys..

[48]  Arne Taube,et al.  A high-order discontinuous Galerkin method with time-accurate local time stepping for the Maxwell equations , 2009 .

[49]  Pierre-Henri Maire,et al.  A high-order one-step sub-cell force-based discretization for cell-centered Lagrangian hydrodynamics on polygonal grids , 2011 .

[50]  S.A.E.G.Falle,et al.  On the Inadmissibility of Non-evolutionary Shocks , 1999 .

[51]  A. Stroud.  Approximate calculation of multiple integrals , 1973 .

[52]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[53]  Mikhail J. Shashkov,et al.  One-step hybrid remapping algorithm for multi-material arbitrary Lagrangian-Eulerian methods , 2012, J. Comput. Phys..

[54]  Claus-Dieter Munz,et al.  A Discontinuous Galerkin Scheme Based on a Space–Time Expansion. I. Inviscid Compressible Flow in One Space Dimension , 2007, J. Sci. Comput..

[55]  Chi-Wang Shu,et al.  Efficient Implementation of Weighted ENO Schemes , 1995 .

[56]  Antonio Baeza,et al.  Adaptation based on interpolation errors for high order mesh refinement methods applied to conservation laws , 2012 .

[57]  Jérôme Breil,et al.  Two-step hybrid conservative remapping for multimaterial arbitrary Lagrangian-Eulerian methods , 2011, J. Comput. Phys..