A Mach-sensitive implicit-explicit scheme adapted to compressible multi-scale flows

Abstract The method presented below focuses on the numerical approximation of the Euler compressible system. It pursues a two-fold objective: being able to accurately follow slow material waves as well as strong shock waves in the context of low Mach number flows. The resulting implicit–explicit(IMEX) fractional step approach leans on a dynamic splitting designed to react to the time fluctuations of the maximal flow Mach number. When the latter rises suddenly, the IMEX scheme, so far driven by a material-wave Courant number, turn into a time-explicit approximate Riemann solver constrained by an acoustic-wave Courant number. It is also possible to enrich the dynamic splitting in order to capture high pressure jumps even when the flow Mach number is low. One-dimensional low Mach number test cases involving single or multiple waves confirm that the present approach is as accurate and efficient as an IMEX Lagrange-Projection method. Besides, numerical results suggest that the stability of the present method holds for any Mach number if the Courant number related to the convective subsystem arising from the splitting is of order unity.

[1]  Jean-Marc Hérard,et al.  A hybrid scheme to compute contact discontinuities in one-dimensional Euler systems , 2002 .

[2]  F. Bouchut.  Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws: and Well-Balanced Schemes for Sources , 2005 .

[3]  Hamed Zakerzadeh.  On the mach-uniformity of the Lagrange-projection scheme , 2016 .

[4]  Mathieu Girardin,et al.  Asymptotic preserving and all-regime Lagrange-Projection like numerical schemes : application to two-phase flows in low mach regime , 2014 .

[5]  Jean-Marc Hérard,et al.  A conservative fractional step method to solve non-isentropic Euler equations , 1997 .

[6]  Jian‐Guo Liu,et al.  An All-Speed Asymptotic-Preserving Method for the Isentropic Euler and Navier-Stokes Equations , 2012 .

[7]  Shi Jin,et al.  Efficient Asymptotic-Preserving (AP) Schemes For Some Multiscale Kinetic Equations , 1999, SIAM J. Sci. Comput..

[8]  P. Degond,et al.  All speed scheme for the low Mach number limit of the isentropic Euler equations , 2009, 0908.1929.

[9]  Frédéric Coquel,et al.  RELAXATION OF FLUID SYSTEMS , 2012 .

[10]  Angus R. Simpson,et al.  Closure of "Large Water-Hammer Pressures for Column Separation in Pipelines" , 1991 .

[11]  Samuel Kokh,et al.  An all-regime Lagrange-Projection like scheme for the gas dynamics equations on unstructured meshes , 2016 .

[12]  S. Clerc,et al.  Numerical Simulation of the Homogeneous Equilibrium Model for Two-Phase Flows , 2000 .

[13]  Hervé Guillard,et al.  Behavior of upwind scheme in the low Mach number limit: III. Preconditioned dissipation for a five equation two phase model , 2008 .

[14]  Raphaël Loubère,et al.  Study of a New Asymptotic Preserving Scheme for the Euler System in the Low Mach Number Limit , 2017, SIAM J. Sci. Comput..

[15]  E. Turkel,et al.  Preconditioned methods for solving the incompressible low speed compressible equations , 1987 .

[16]  Stéphane Dellacherie,et al.  Construction of modified Godunov-type schemes accurate at any Mach number for the compressible Euler system , 2016 .

[17]  Samuel Kokh,et al.  An all-regime Lagrange-Projection like scheme for 2D homogeneous models for two-phase flows on unstructured meshes , 2017, J. Comput. Phys..

[18]  T. Hou,et al.  Why nonconservative schemes converge to wrong solutions: error analysis , 1994 .

[19]  P. Comte,et al.  Self-Adaptive Newton-based iteration strategy for the LES of turbulent multi-scale flows , 2014 .

[20]  R. Klein.  Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics , 1995 .

[21]  S. Noelle,et al.  A note on the stability of implicit-explicit flux-splittings for stiff systems of hyperbolic conservation laws , 2018 .

[22]  G. Whitham.  Linear and non linear waves , 1974 .

[23]  H. Guillard,et al.  On the behaviour of upwind schemes in the low Mach number limit , 1999 .

[24]  G. Mcguire,et al.  Explicit-implicit schemes for the numerical solution of nonlinear hyperbolic systems , 1975 .

[25]  G. Mcguire,et al.  A class of implicit, second-order accurate, dissipative schemes for solving systems of conservation laws , 1974 .

[26]  François Bouchut,et al.  Entropy satisfying flux vector splittings and kinetic BGK models , 2003, Numerische Mathematik.

[27]  Graham V. Candler,et al.  A parallel implicit method for the direct numerical simulation of wall-bounded compressible turbulence , 2006, J. Comput. Phys..

[28]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

[29]  Rémy Baraille,et al.  Une version à pas fractionnaires du schéma de Godunov pour l'hydrodynamique , 1992 .

[30]  Jochen Schütz,et al.  Flux Splitting for Stiff Equations: A Notion on Stability , 2014, J. Sci. Comput..

[31]  Frédéric Coquel,et al.  Entropy-satisfying relaxation method with large time-steps for Euler IBVPs , 2010, Math. Comput..

[32]  Mohamed Salah Ghidaoui,et al.  A Review of Water Hammer Theory and Practice , 2005 .

[33]  A. Lerat.  Implicit methods of second-order accuracy for the Euler equations , 1983 .

[34]  Claus-Dieter Munz,et al.  A Weakly Asymptotic Preserving Low Mach Number Scheme for the Euler Equations of Gas Dynamics , 2014, SIAM J. Sci. Comput..

[35]  Lorenzo Allievi,et al.  Teoria generale del moto perturbato dell'acqua nei tubi in pressione (colpo d'ariete) memoria dell'ing. L. B. Allievi , 1903 .

[36]  P. Goorjian,et al.  Implicit Finite-Difference Computations of Unsteady Transonic Flows about Airfoils , 1977 .

[37]  H. Guillard,et al.  On the behavior of upwind schemes in the low Mach number limit: II. Godunov type schemes , 2004 .

[38]  Pascal Galon,et al.  A Mach-Sensitive Splitting Approach for Euler-like Systems , 2018 .