High order cell-centered Lagrangian-type finite volume schemes with time-accurate local time stepping on unstructured triangular meshes

We present a novel cell-centered direct Arbitrary-Lagrangian-Eulerian (ALE) finite volume scheme on unstructured triangular meshes that is high order accurate in space and time and that also allows for time-accurate local time stepping (LTS). It extends our previous investigations on high order Lagrangian finite volume schemes with LTS carried out in 46] in one space dimension. The new scheme uses the following basic ingredients: a high order WENO reconstruction in space on unstructured meshes, an element-local high-order accurate space-time Galerkin predictor that performs the time evolution of the reconstructed polynomials within each element, the computation of numerical ALE fluxes at the moving element interfaces through approximate Riemann solvers, and a one-step finite volume scheme for the time update which is directly based on the integral form of the conservation equations in space-time. The inclusion of the LTS algorithm requires a number of crucial extensions, such as a proper scheduling criterion for the time update of each element and for each node; a virtual projection of the elements contained in the reconstruction stencils of the element that has to perform the WENO reconstruction; and the proper computation of the fluxes through the space-time boundary surfaces that will inevitably contain hanging nodes in time due to the LTS algorithm.We have validated our new unstructured Lagrangian LTS approach over a wide sample of test cases solving the Euler equations of compressible gas dynamics in two space dimensions, including shock tube problems, cylindrical explosion problems, as well as specific tests typically adopted in Lagrangian calculations, such as the Kidder, the Saltzman and the Sedov problem. When compared to the traditional global time stepping (GTS) method, the newly proposed LTS algorithm allows to reduce the number of element updates in a given simulation by a factor that may depend on the complexity of the dynamics, but which can be as large as ~4.7. Finally, we have also shown the improvement in terms of computational efficiency in a representative test for the special relativistic magnetohydrodynamics (RMHD) equations.

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

[2]  Rémi Abgrall,et al.  Multidimensional HLLC Riemann solver for unstructured meshes - With application to Euler and MHD flows , 2014, J. Comput. Phys..

[3]  Dimitris Drikakis,et al.  WENO schemes for mixed-elementunstructured meshes , 2010 .

[4]  Vladimir A. Titarev,et al.  WENO schemes on arbitrary mixed-element unstructured meshes in three space dimensions , 2011, J. Comput. Phys..

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

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

[7]  INFN,et al.  The exact solution of the Riemann problem in relativistic magnetohydrodynamics , 2005, Journal of Fluid Mechanics.

[8]  Michael Dumbser,et al.  A new class of Moving-Least-Squares WENO-SPH schemes , 2014, J. Comput. Phys..

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

[10]  Michael Dumbser,et al.  A high order special relativistic hydrodynamic and magnetohydrodynamic code with space-time adaptive mesh refinement , 2013, Comput. Phys. Commun..

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

[12]  R. Menikoff,et al.  The Riemann problem for fluid flow of real materials , 1989 .

[13]  P. Frederickson,et al.  Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction , 1990 .

[14]  Bernardo Cockburn.  Discontinuous Galerkin methods , 2003 .

[15]  G. Stelling,et al.  Semi‐implicit subgrid modelling of three‐dimensional free‐surface flows , 2011 .

[16]  P. Colella,et al.  Local adaptive mesh refinement for shock hydrodynamics , 1989 .

[17]  Xijun Yu,et al.  The cell-centered discontinuous Galerkin method for Lagrangian compressible Euler equations in two-dimensions , 2014 .

[18]  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..

[19]  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.

[20]  Siegfried Müller,et al.  The Riemann Problem for the Euler Equations with Nonconvex and Nonsmooth Equation of State: Construction of Wave Curves , 2006, SIAM J. Sci. Comput..

[21]  Chi-Wang Shu,et al.  Discontinuous Galerkin Methods: Theory, Computation and Applications , 2011 .

[22]  Pierre-Henri Maire,et al.  Multi-scale Godunov-type method for cell-centered discrete Lagrangian hydrodynamics , 2009, J. Comput. Phys..

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

[24]  Stéphane Clain,et al.  A high-order finite volume method for systems of conservation laws - Multi-dimensional Optimal Order Detection (MOOD) , 2011, J. Comput. Phys..

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

[26]  Phillip Colella,et al.  Efficient Solution Algorithms for the Riemann Problem for Real Gases , 1985 .

[27]  Chi-Wang Shu,et al.  Monotonicity Preserving Weighted Essentially Non-oscillatory Schemes with Increasingly High Order of Accuracy , 2000 .

[28]  Eleuterio F. Toro,et al.  Derivative Riemann solvers for systems of conservation laws and ADER methods , 2006, J. Comput. Phys..

[29]  Jérôme Breil,et al.  A multi-material ReALE method with MOF interface reconstruction , 2013 .

[30]  Stéphane Clain,et al.  Improved detection criteria for the multi-dimensional optimal order detection (MOOD) on unstructured meshes with very high-order polynomials , 2012 .

[31]  Dinshaw S. Balsara.  A two-dimensional HLLC Riemann solver for conservation laws: Application to Euler and magnetohydrodynamic flows , 2012, J. Comput. Phys..

[32]  Rémi Abgrall,et al.  A discontinuous Galerkin discretization for solving the two-dimensional gas dynamics equations written under total Lagrangian formulation on general unstructured grids , 2014, J. Comput. Phys..

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

[34]  Rémi Abgrall,et al.  Cell-centered discontinuous Galerkin discretizations for two-dimensional scalar conservation laws on unstructured grids and for one-dimensional Lagrangian hydrodynamics , 2011 .

[35]  Armin Iske,et al.  Adaptive ADER Methods Using Kernel-Based Polyharmonic Spline WENO Reconstruction , 2010, SIAM J. Sci. Comput..

[36]  François Vilar,et al.  Cell-centered discontinuous Galerkin discretization for two-dimensional Lagrangian hydrodynamics , 2012 .

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

[38]  Michael Dumbser,et al.  High‐order ADER‐WENO ALE schemes on unstructured triangular meshes—application of several node solvers to hydrodynamics and magnetohydrodynamics , 2013, 1310.7256.

[39]  Dinshaw S. Balsara,et al.  Total Variation Diminishing Scheme for Relativistic Magnetohydrodynamics , 2001 .

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

[41]  P. Londrillo,et al.  An efficient shock-capturing central-type scheme for multidimensional relativistic flows. II. Magnetohydrodynamics , 2002 .

[42]  Tzanio V. Kolev,et al.  High-order curvilinear finite elements for axisymmetric Lagrangian hydrodynamics , 2013 .

[43]  Michael Dumbser,et al.  An Efficient Quadrature-Free Formulation for High Order Arbitrary-Lagrangian–Eulerian ADER-WENO Finite Volume Schemes on Unstructured Meshes , 2016, J. Sci. Comput..

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

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

[46]  William J. Rider,et al.  Multi-material pressure relaxation methods for Lagrangian hydrodynamics , 2013 .

[47]  Vincenzo Casulli,et al.  A semi‐implicit numerical method for the free‐surface Navier–Stokes equations , 2014 .

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

[49]  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..

[50]  James M. Stone,et al.  A hybrid Godunov method for radiation hydrodynamics , 2010, J. Comput. Phys..

[51]  S. K. Trehan,et al.  Plasma oscillations (I) , 1960 .

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

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

[54]  Bruno Després,et al.  Symmetrization of Lagrangian gas dynamic in dimension two and multidimensional solvers , 2003 .

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

[56]  July Stuttgart.  A High Order Sharp{Interface Method with Local Time Stepping for Compressible Multiphase Flows , 2011 .

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

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

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

[60]  S. Komissarov,et al.  A Godunov-type scheme for relativistic magnetohydrodynamics , 1999 .

[61]  Michael Dumbser,et al.  Arbitrary high order finite volume schemes for linear wave propagation , 2006 .

[62]  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..

[63]  Raphaël Loubère,et al.  3D staggered Lagrangian hydrodynamics scheme with cell‐centered Riemann solver‐based artificial viscosity , 2013 .

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

[65]  Guang-Shan Jiang,et al.  A High-Order WENO Finite Difference Scheme for the Equations of Ideal Magnetohydrodynamics , 1999 .

[66]  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.

[67]  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..

[68]  Mikhail Shashkov,et al.  A comparative study of multimaterial Lagrangian and Eulerian methods with pressure relaxation , 2013 .

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

[70]  Dinshaw S. Balsara.  Multidimensional HLLE Riemann solver: Application to Euler and magnetohydrodynamic flows , 2010, J. Comput. Phys..

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

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

[73]  Moshe Dubiner.  Spectral methods on triangles and other domains , 1991 .

[74]  Michael Dumbser,et al.  Arbitrary-Lagrangian–Eulerian ADER–WENO finite volume schemes with time-accurate local time stepping for hyperbolic conservation laws , 2014, 1402.6897.

[75]  Jaromír Horácek,et al.  Simulation of compressible viscous flow in time-dependent domains , 2013, Appl. Math. Comput..

[76]  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..

[77]  Vít Dolejší,et al.  A semi-implicit discontinuous Galerkin finite element method for the numerical solution of inviscid compressible flow , 2004 .

[78]  Samuel Paolucci,et al.  Accurate Spatial Resolution Estimates for Reactive Supersonic Flow with Detailed Chemistry , 2005 .

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

[80]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[81]  G. Karniadakis,et al.  Spectral/hp Element Methods for CFD , 1999 .

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

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

[84]  Yong-Tao Zhang,et al.  Third Order WENO Scheme on Three Dimensional Tetrahedral Meshes , 2008 .

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

[86]  Miloslav Feistauer,et al.  Numerical analysis of flow-induced nonlinear vibrations of an airfoil with three degrees of freedom , 2011 .

[87]  Mikhail Shashkov,et al.  Exploration of new limiter schemes for stress tensors in Lagrangian and ALE hydrocodes , 2013 .

[88]  Jérôme Breil,et al.  A second‐order cell‐centered Lagrangian scheme for two‐dimensional compressible flow problems , 2008 .

[89]  R. Kidder,et al.  Laser-driven compression of hollow shells: power requirements and stability limitations , 1976 .

[90]  C. Ollivier-Gooch,et al.  A high-order-accurate unstructured mesh finite-volume scheme for the advection-diffusion equation , 2002 .

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

[92]  Chau-Wen Tseng,et al.  A message passing benchmark for unbalanced applications , 2008, Simul. Model. Pract. Theory.

[93]  Vincenzo Casulli,et al.  A SEMI-IMPLICIT FINITE DIFFERENCE METHOD FOR NON-HYDROSTATIC, FREE-SURFACE FLOWS , 1999 .

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

[95]  John K. Dukowicz,et al.  Vorticity errors in multidimensional Lagrangian codes , 1992 .

[96]  Vít Dolejší,et al.  Analysis of semi-implicit DGFEM for nonlinear convection–diffusion problems on nonconforming meshes ☆ , 2007 .

[97]  V. Dolejší,et al.  Semi-Implicit Interior Penalty Discontinuous Galerkin Methods for Viscous Compressible Flows , 2008 .

[98]  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..

[99]  Armin Iske,et al.  ADER schemes on adaptive triangular meshes for scalar conservation laws , 2005 .

[100]  Bruno Després,et al.  A new exceptional points method with application to cell-centered Lagrangian schemes and curved meshes , 2012, J. Comput. Phys..

[101]  Ronald Fedkiw,et al.  High Accuracy Numerical Methods for Thermally Perfect Gas Flows with Chemistry , 1997 .

[102]  Angelo Marcello Anile,et al.  Relativistic fluids and magneto-fluids , 2005 .

[103]  Michael Dumbser,et al.  A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes , 2008, J. Comput. Phys..

[104]  B R U N O G I A C O M A Z Z O,et al.  Under consideration for publication in J. Fluid Mech. 1 The Exact Solution of the Riemann Problem in Relativistic MHD , 2008 .

[105]  Veselin Dobrev,et al.  Curvilinear finite elements for Lagrangian hydrodynamics , 2011 .

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

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

[108]  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 .

[109]  Tzanio V. Kolev,et al.  High-Order Curvilinear Finite Element Methods for Lagrangian Hydrodynamics , 2012, SIAM J. Sci. Comput..

[110]  N. Bucciantini,et al.  An efficient shock-capturing central-type scheme for multidimensional relativistic flows , 2002 .

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

[112]  Michael Dumbser,et al.  Efficient implementation of high order unstructured WENO schemes for cavitating flows , 2013 .

[113]  L. Rezzolla,et al.  General relativistic radiation hydrodynamics of accretion flows - I. Bondi-Hoyle accretion , 2011, 1105.5615.

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

[115]  W. Marsden.  I and J , 2012 .

[116]  M. Berger,et al.  Adaptive mesh refinement for hyperbolic partial differential equations , 1982 .

[117]  Michael Dumbser,et al.  A New Family of High Order Unstructured MOOD and ADER Finite Volume Schemes for Multidimensional Systems of Hyperbolic Conservation Laws , 2014 .

[118]  Michael Dumbser,et al.  Lagrangian ADER-WENO finite volume schemes on unstructured triangular meshes based on genuinely multidimensional HLL Riemann solvers , 2013, J. Comput. Phys..

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