Invariant domains preserving ALE approximation of hyperbolic systems with continuous finite elements

A conservative invariant domain preserving Arbitrary Lagrangian Eulerian method for solving nonlinear hyperbolic systems is introduced. The method is explicit in time, works with continuous finite elements and is first-order accurate in space. One originality of the present work is that the artificial viscosity is unambiguously defined irrespective of the mesh geometry/anisotropy and does not depend on any ad hoc parameter. The proposed method is meant to be a stepping stone for the construction of higher-order methods in space by using appropriate limitation techniques.

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

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

[3]  Mikhail Shashkov,et al.  A tensor artificial viscosity using a mimetic finite difference algorithm , 2001 .

[4]  M. N. Spijker,et al.  An extension and analysis of the Shu-Osher representation of Runge-Kutta methods , 2004, Math. Comput..

[5]  Paul Glaister,et al.  Generation of Arbitrary Lagrangian–Eulerian (ALE) velocities, based on monitor functions, for the solution of compressible fluid equations , 2005 .

[6]  Larry L. Schumaker,et al.  Spline Functions on Triangulations: Triangulations and Quadrangulations , 2007 .

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

[8]  Raphaël Loubère,et al.  "Curl-q": A vorticity damping artificial viscosity for essentially irrotational Lagrangian hydrodynamics calculations , 2006, J. Comput. Phys..

[9]  Zhi Yang,et al.  Unstructured Dynamic Meshes with Higher-order Time Integration Schemes for the Unsteady Navier-Stokes Equations , 2005 .

[10]  Chi-Wang Shu,et al.  Positivity-preserving Lagrangian scheme for multi-material compressible flow , 2014, J. Comput. Phys..

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

[12]  W. Bangerth,et al.  deal.II—A general-purpose object-oriented finite element library , 2007, TOMS.

[13]  A. Bressan.  Hyperbolic Systems of Conservation Laws , 1999 .

[14]  Inmaculada Higueras,et al.  Representations of Runge-Kutta Methods and Strong Stability Preserving Methods , 2005, SIAM J. Numer. Anal..

[15]  Chi-Wang Shu,et al.  High Order Strong Stability Preserving Time Discretizations , 2009, J. Sci. Comput..

[16]  P. Lax.  Weak solutions of nonlinear hyperbolic equations and their numerical computation , 1954 .

[17]  J. Guermond,et al.  Fast estimation of the maximum wave speed in the Riemann problem for the Euler equations , 2015 .

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

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

[20]  Gui-Qiang G. Chen.  Euler Equations and Related Hyperbolic Conservation Laws , 2005 .

[21]  Konstantin Lipnikov,et al.  A framework for developing a mimetic tensor artificial viscosity for Lagrangian hydrocodes on arbitrary polygonal meshes , 2010, J. Comput. Phys..

[22]  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..

[23]  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 .

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

[25]  Lucia Gastaldi,et al.  A priori error estimates for the Arbitrary Lagrangian Eulerian formulation with finite elements , 2001, J. Num. Math..

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

[27]  C. Dafermos.  Hyberbolic Conservation Laws in Continuum Physics , 2000 .

[28]  Mikhail Shashkov,et al.  Multi-Scale Lagrangian Shock Hydrodynamics on Q1/P0 Finite Elements: Theoretical Framework and Two-dimensional Computations. , 2008 .

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

[30]  Tzanio V. Kolev,et al.  A tensor artificial viscosity using a finite element approach , 2009, J. Comput. Phys..

[31]  D. Hoff.  Invariant regions for systems of conservation laws , 1985 .

[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]  Mark Ainsworth,et al.  Pyramid Algorithms for Bernstein-Bézier Finite Elements of High, Nonuniform Order in Any Dimension , 2014, SIAM J. Sci. Comput..

[34]  P. Lax.  Hyperbolic systems of conservation laws , 2006 .

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

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