High-order staggered schemes for compressible hydrodynamics. Weak consistency and numerical validation

Abstract Staggered grids schemes, formulated in internal energy, are commonly used for CFD applications in industrial context. Here, we prove the consistency of a class of high-order Lagrange-Remap staggered schemes for solving the Euler equations in 1D and 2D on Cartesian grids. The main result of the paper is that using an a posteriori internal energy corrector, the Lagrangian schemes are proved to be conservative in mass, momentum and total energy and to be weakly consistent with the 1D Lagrangian formulation of the Euler equations. Extension in 2D is done using directional splitting methods and face-staggering. Numerical examples in both 1D and 2D illustrate the accuracy, the convergence and the robustness of the schemes.

[1]  H. Huynh,et al.  Accurate Monotonicity-Preserving Schemes with Runge-Kutta Time Stepping , 1997 .

[2]  P. Lax,et al.  Systems of conservation laws , 1960 .

[3]  Raphaèle Herbin,et al.  Consistent segregated staggered schemes with explicit steps for the isentropic and full Euler equations , 2018 .

[4]  Raphaèle Herbin,et al.  EXPLICIT STAGGERED SCHEMES FOR THE COMPRESSIBLE EULER EQUATIONS , 2013 .

[5]  K. R. Trigger,et al.  NUMERICAL SOLUTION OF THE ONE-DIMENSIONAL LAGRANGIAN HYDRODYNAMIC EQUATIONS , 1961 .

[6]  Raphaël Loubère,et al.  The internal consistency, stability, and accuracy of the discrete, compatible formulation of Lagrangian hydrodynamics , 2006, J. Comput. Phys..

[7]  G. Quispel,et al.  Acta Numerica 2002: Splitting methods , 2002 .

[8]  Gautier Dakin,et al.  High-order accurate Lagrange-remap hydrodynamic schemes on staggered Cartesian grids , 2016 .

[9]  Hervé Jourdren,et al.  Dissipative issue of high-order shock capturing schemes with non-convex equations of state , 2009, J. Comput. Phys..

[10]  I. N. Sneddon,et al.  Systems of Conservation Laws 1: Hyperbolicity, Entropies, Shock Waves , 1999 .

[11]  M. Wolff.  Mathematical and numerical analysis of the resistive magnetohydrodynamics system with self-generated magnetic field terms , 2011 .

[12]  Rajan Arora,et al.  Propagation of strong shock waves in a non-ideal gas , 2019, Acta Astronautica.

[13]  Richard Liska,et al.  Comparison of Several Difference Schemes on 1D and 2D Test Problems for the Euler Equations , 2003, SIAM J. Sci. Comput..

[14]  M. Shashkov,et al.  The Construction of Compatible Hydrodynamics Algorithms Utilizing Conservation of Total Energy , 1998 .

[15]  Akio Arakawa,et al.  Computational Design of the Basic Dynamical Processes of the UCLA General Circulation Model , 1977 .

[16]  H. Yoshida.  Construction of higher order symplectic integrators , 1990 .

[17]  R. McLachlan,et al.  The accuracy of symplectic integrators , 1992 .

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

[19]  W. Cabot,et al.  A high-wavenumber viscosity for high-resolution numerical methods , 2004 .

[20]  P. Raviart,et al.  Numerical Approximation of Hyperbolic Systems of Conservation Laws , 1996, Applied Mathematical Sciences.

[21]  A. A. Samarskii,et al.  Completely conservative difference schemes , 1969 .

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

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

[24]  G. Taylor.  The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I , 1950, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[25]  Ben Thornber,et al.  Accuracy of high‐order density‐based compressible methods in low Mach vortical flows , 2014 .

[26]  Stéphane Jaouen,et al.  High-order dimensionally split Lagrange-remap schemes for compressible hydrodynamics , 2010 .

[27]  G. Sod.  A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws , 1978 .

[28]  J. Dormand,et al.  A family of embedded Runge-Kutta formulae , 1980 .

[29]  R. D. Richtmyer,et al.  A Method for the Numerical Calculation of Hydrodynamic Shocks , 1950 .

[30]  Kinetic energy control in the MAC discretization of compressible Navier-Stokes equations , 2010 .

[31]  Bruno Després.  Weak consistency of the cell-centered Lagrangian GLACE scheme on general meshes in any dimension , 2010 .

[32]  W. F. Noh.  Numerical methods in hydrodynamic calculations , 1976 .

[33]  Alexandra Claisse,et al.  Energy preservation and entropy in Lagrangian space- and time-staggered hydrodynamic schemes , 2016, J. Comput. Phys..

[34]  T. Gallouët,et al.  Kinetic energy control in explicit Finite Volume discretizations of the incompressible and compressible Navier-Stokes equations , 2010 .

[35]  P. Lax.  Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves , 1987 .

[36]  Neil D. Sandham,et al.  Low-Dissipative High-Order Shock-Capturing Methods Using Characteristic-Based Filters , 1999 .

[37]  Chi-Wang Shu,et al.  Total variation diminishing Runge-Kutta schemes , 1998, Math. Comput..

[38]  R. Herbin,et al.  Consistent explicit staggered schemes for compressible flows Part II: the Euler equation , 2013 .

[39]  Robert I. McLachlan,et al.  On the Numerical Integration of Ordinary Differential Equations by Symmetric Composition Methods , 1995, SIAM J. Sci. Comput..

[40]  Chi-Wang Shu,et al.  Strong Stability-Preserving High-Order Time Discretization Methods , 2001, SIAM Rev..

[41]  Hervé Guillard,et al.  On the Behaviour of Upwind Schemes in the Low Mach Number Limit: A Review , 2017 .

[42]  Ben Thornber,et al.  Large-eddy simulation of multi-component compressible turbulent flows using high resolution methods , 2008 .

[43]  J. Strutt.  Scientific Papers: Investigation of the Character of the Equilibrium of an Incompressible Heavy Fluid of Variable Density , 2009 .

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