Second-order invariant domain preserving ALE approximation of hyperbolic systems

Abstract In this paper we introduce an invariant domain preserving arbitrary Lagrangian Eulerian method for solving hyperbolic systems. The time stepping is explicit and the approximation in space is done with continuous finite elements. The method is second-order in space and made invariant domain preserving by using a newly introduced convex limiting technique.

[1]  Guglielmo Scovazzi,et al.  Stabilized shock hydrodynamics: II. Design and physical interpretation of the SUPG operator for Lagrangian computations☆ , 2007 .

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

[3]  Charbel Farhat,et al.  On the significance of the geometric conservation law for flow computations on moving meshes , 2000 .

[4]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[5]  A. Jameson,et al.  Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes , 1981 .

[6]  Jean-Luc Guermond,et al.  Invariant Domains and Second-Order Continuous Finite Element Approximation for Scalar Conservation Equations , 2017, SIAM J. Numer. Anal..

[7]  H. Frid.  Maps of Convex Sets and Invariant Regions¶for Finite-Difference Systems¶of Conservation Laws , 2001 .

[8]  Mikhail J. Shashkov,et al.  Adaptive reconnection-based arbitrary Lagrangian Eulerian method , 2015, J. Comput. Phys..

[9]  Jean-Luc Guermond,et al.  Invariant domain preserving discretization-independent schemes and convex limiting for hyperbolic systems , 2018, Computer Methods in Applied Mechanics and Engineering.

[10]  P. Thomas,et al.  Geometric Conservation Law and Its Application to Flow Computations on Moving Grids , 1979 .

[11]  Bojan Popov,et al.  Adaptive Semidiscrete Central-Upwind Schemes for Nonconvex Hyperbolic Conservation Laws , 2007, SIAM J. Sci. Comput..

[12]  Ami Harten,et al.  Convex Entropies and Hyperbolicity for General Euler Equations , 1998 .

[13]  N. Dyn,et al.  A butterfly subdivision scheme for surface interpolation with tension control , 1990, TOGS.

[14]  Jean-Luc Guermond,et al.  Well-Balanced Second-Order Finite Element Approximation of the Shallow Water Equations with Friction , 2018, SIAM J. Sci. Comput..

[15]  Yong Yang,et al.  Invariant Domains Preserving Arbitrary Lagrangian Eulerian Approximation of Hyperbolic Systems with Continuous Finite Elements , 2017, SIAM J. Sci. Comput..

[16]  A. Huerta,et al.  Arbitrary Lagrangian–Eulerian Methods , 2004 .

[17]  Jean-Luc Guermond,et al.  Second-Order Invariant Domain Preserving Approximation of the Euler Equations Using Convex Limiting , 2017, SIAM J. Sci. Comput..

[18]  Raphaël Loubère,et al.  ReALE: A reconnection-based arbitrary-Lagrangian-Eulerian method , 2010, J. Comput. Phys..

[19]  Bojan Popov,et al.  Fast estimation from above of the maximum wave speed in the Riemann problem for the Euler equations , 2015, J. Comput. Phys..

[20]  Benoît Perthame,et al.  Maximum principle on the entropy and second-order kinetic schemes , 1994 .

[21]  Rémi Abgrall,et al.  Staggered Grid Residual Distribution Scheme for Lagrangian Hydrodynamics , 2017, SIAM J. Sci. Comput..

[22]  Erik Burman,et al.  On nonlinear artificial viscosity, discrete maximum principle and hyperbolic conservation laws , 2007 .

[23]  S. Zalesak.  Fully multidimensional flux-corrected transport algorithms for fluids , 1979 .

[24]  Jean-Luc Guermond,et al.  Entropy–viscosity method for the single material Euler equations in Lagrangian frame , 2016 .

[25]  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..

[26]  Charbel Farhat,et al.  The discrete geometric conservation law and the nonlinear stability of ALE schemes for the solution of flow problems on moving grids , 2001 .

[27]  Guglielmo Scovazzi,et al.  Galilean invariance and stabilized methods for compressible flows , 2007 .

[28]  C. C. Long,et al.  Isogeometric analysis of Lagrangian hydrodynamics: Axisymmetric formulation in the rz-cylindrical coordinates , 2014, J. Comput. Phys..

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

[30]  Mikhail Shashkov,et al.  Formulations of Artificial Viscosity for Multi-dimensional Shock Wave Computations , 1998 .

[31]  Michael Dumbser,et al.  Direct Arbitrary-Lagrangian-Eulerian ADER-MOOD finite volume schemes for multidimensional hyperbolic conservation laws , 2015, J. Comput. Phys..

[32]  Tzanio V. Kolev,et al.  High-Order Multi-Material ALE Hydrodynamics , 2018, SIAM J. Sci. Comput..

[33]  Michael Dumbser,et al.  A second-order cell-centered Lagrangian ADER-MOOD finite volume scheme on multidimensional unstructured meshes for hydrodynamics , 2018, J. Comput. Phys..

[34]  Bojan Popov,et al.  Invariant Domains and First-Order Continuous Finite Element Approximation for Hyperbolic Systems , 2015, SIAM J. Numer. Anal..

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

[36]  Larry L. Schumaker,et al.  Spline functions on triangulations , 2007, Encyclopedia of mathematics and its applications.

[37]  Jean-Luc Guermond,et al.  Entropy viscosity method for nonlinear conservation laws , 2011, J. Comput. Phys..

[38]  Jean-Luc Guermond,et al.  Weighting the Edge Stabilization , 2013, SIAM J. Numer. Anal..

[39]  W. F. Noh.  Errors for calculations of strong shocks using an artificial viscosity and artificial heat flux , 1985 .

[40]  Jay P. Boris,et al.  Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works , 1973 .

[41]  W. F. Noh,et al.  CEL: A TIME-DEPENDENT, TWO-SPACE-DIMENSIONAL, COUPLED EULERIAN-LAGRANGE CODE , 1963 .

[42]  D. Serre,et al.  About the relative entropy method for hyperbolic systems of conservation laws , 2014 .

[43]  J. Kraaijevanger.  Contractivity of Runge-Kutta methods , 1991 .

[44]  A. J. Barlow,et al.  A compatible finite element multi‐material ALE hydrodynamics algorithm , 2008 .

[45]  Antony Jameson,et al.  Origins and Further Development of the Jameson–Schmidt–Turkel Scheme , 2017 .

[46]  P. Woodward,et al.  The numerical simulation of two-dimensional fluid flow with strong shocks , 1984 .