In the present work, we show that the Implicit-Explicit Lagrange-projection scheme applied to the isentropic Euler equations, presented in Coquel et al.’s paper (Math. Comp. 79 (2010) 1493–1533), is asymptotic preserving regarding the Mach number, ., it is asymptotically stable in -norm with unrestrictive CFL condition for all-Mach flows, and asymptotically consistent which means that it gives a consistent discretization to the incompressible Euler equations in the limit, ., it preserves the incompressible limit as to satisfy the -free condition and the analogues of continuous-level asymptotic expansion for the density. This consistency analysis has been done formally as well as rigorously. Moreover, we prove that the scheme is positivity-preserving and entropy-admissible under some Mach-uniform restrictions. The analysis is similar to what has been presented in the original paper, but with the emphasis on the uniformity regarding the Mach number, which is crucial for a scheme to be useful in the low-Mach regime. We then extend the modified (but similar) analysis to the shallow water equations with topography and get similar stability and consistency results.
Accepté le :
DOI : 10.1051/m2an/2016064
Mots clés : All-Mach number scheme, Lagrange-projection scheme, asymptotic preserving scheme, stability analysis
@article{M2AN_2017__51_4_1343_0, author = {Zakerzadeh, Hamed}, title = {On the mach-uniformity of the {Lagrange-projection} scheme}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {1343--1366}, publisher = {EDP-Sciences}, volume = {51}, number = {4}, year = {2017}, doi = {10.1051/m2an/2016064}, mrnumber = {3702416}, zbl = {06790780}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2016064/} }
TY - JOUR AU - Zakerzadeh, Hamed TI - On the mach-uniformity of the Lagrange-projection scheme JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 1343 EP - 1366 VL - 51 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2016064/ DO - 10.1051/m2an/2016064 LA - en ID - M2AN_2017__51_4_1343_0 ER -
%0 Journal Article %A Zakerzadeh, Hamed %T On the mach-uniformity of the Lagrange-projection scheme %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 1343-1366 %V 51 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2016064/ %R 10.1051/m2an/2016064 %G en %F M2AN_2017__51_4_1343_0
Zakerzadeh, Hamed. On the mach-uniformity of the Lagrange-projection scheme. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 4, pp. 1343-1366. doi : 10.1051/m2an/2016064. http://www.numdam.org/articles/10.1051/m2an/2016064/
Splitting methods for low Mach number Euler and Navier–Stokes equations. Comput. Fluids 17 (1989) 1–12. | DOI | Zbl
, and ,G. Bispen, IMEX finite volume methods for the shallow water equations. Ph.D. thesis, Johannes Gutenberg-Universität (2015).
IMEX large time step finite volume methods for low Froude number shallow water flows. Commun. Comput. Phys. 16 (2014) 307–347. | DOI | MR | Zbl
, , and ,Gravity driven shallow water models for arbitrary topography. Commun. Math. Sci. 2 (2004) 359–389. | DOI | MR | Zbl
and ,F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws: And well-balanced schemes for sources. Springer Science & Business Media (2004). | MR | Zbl
Relaxation approximation of the Euler equations. J. Math. Anal. Appl. 348 (2008) 872–893. | DOI | MR | Zbl
and ,Large time step and asymptotic preserving numerical schemes for the gas dynamics equations with source terms. SIAM J. Sci. Comput. 35 (2013) A2874–A2902. | DOI | MR | Zbl
, and ,An all-regime Lagrange-projection like scheme for the gas dynamics equations on unstructured meshes. Commun. Comput. Phys. 20 (2016) 188–233. | DOI | MR | Zbl
, and ,C. Chalons, P. Kestener, S. Koch and M. Stauffert, A large time-step and well-balanced Lagrange-projection type scheme for the shallow water equations, Preprint (2016). | arXiv | MR
On the Eulerian large eddy simulation of disperse phase flows: An asymptotic preserving scheme for small Stokes number flows. Multiscale Model. Simul. 13 (2015) 291–315. | DOI | MR | Zbl
, and ,Hyperbolic conservation laws with stiff relaxation terms and entropy. Commun. Pure Appl. Math. 47 (1994) 787–830. | DOI | MR | Zbl
, and ,F. Coquel, E. Godlewski and N. Seguin, Regularization and relaxation tools for interface coupling. In Proc. of XX Congresso de Ecuaciones Differenciales Y Aplicaicones, CEDYA (2007).
Entropy-satisfying relaxation method with large time-steps for Euler IBVPs. Math. Comput. 79 (2010) 1493–1533. | DOI | MR | Zbl
, , and ,An asymptotic-preserving all-speed scheme for the Euler and Navier–Stokes equations. J. Comput. Phys. 231 (2012) 5685–5704. | DOI | MR | Zbl
, and ,Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74 (1995) 483–548. | MR | Zbl
, and ,Analysis of an asymptotic preserving scheme for the Euler–Poisson system in the quasineutral limit. SIAM J. Numer. Anal. 46 (2008) 1298–1322. | DOI | MR | Zbl
, and ,All speed scheme for the low Mach number limit of the isentropic Euler equation. Commun. Comput. Phys. 10 (2011) 1–31. | DOI | MR | Zbl
and ,Analysis of Godunov type schemes applied to the compressible Euler system at low Mach number. J. Comput. Phys. 229 (2010) 978–1016. | DOI | MR | Zbl
,S. Dellacherie, J. Jung, P. Omnes and P.-A. Raviart, Construction of modified Godunov type schemes accurate at any Mach number for the compressible Euler system. submitted, hal-00776629, November 2015. | MR
The influence of cell geometry on the Godunov scheme applied to the linear wave equation. J. Comput. Phys. 229 (2010) 5315–5338. | DOI | MR | Zbl
, and ,J. Donea, A. Huerta, J.-Ph. Ponthot and A. Rodríguez-Ferran, Arbitrary Lagrangian–Eulerian methods. John Wiley & Sons, Ltd (2004).
E. Godlewski and P.-A. Raviart, Numerical approximation of hyperbolic systems of conservation laws. Vol. 118. Springer (1996). | MR | Zbl
R.M. Gray, Toeplitz and circulant matrices: A review. Now Publishers Inc. (2006). | Zbl
Oscillatory perturbations of the Navier–Stokes equations. J. Math. Pures Appl. 76 (1997) 477–498. | DOI | MR | Zbl
,On the behavior of upwind schemes in the low Mach number limit: II. Godunov type schemes. Comput. Fluids 33 (2004) 655–675. | DOI | Zbl
and ,On the behaviour of upwind schemes in the low Mach number limit. Comput. Fluids 28 (1999) 63–86. | DOI | MR | Zbl
and ,An all-speed asymptotic-preserving method for the isentropic Euler and Navier–Stokes equations. Commun. Comput. Phys. 12 (2012) 955–980. | DOI | MR | Zbl
, and ,H. Holden, K.H. Karlsen, K.-A. Lie and N.H. Risebro, Splitting methods for partial differential equations with rough solutions. European Math. Soc. Publishing House (2010). | MR | Zbl
R.A. Horn and C.R. Johnson, Matrix analysis. Cambridge University Press, New York (1986). | MR | Zbl
Runge–Kutta methods for hyperbolic conservation laws with stiff relaxation terms. J. Comput. Phys. 122 (1995) 51–67. | DOI | MR | Zbl
,Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21 (1999) 441–454. | DOI | MR | Zbl
,S. Jin, Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: A review. Lecture Notes for Summer School on Methods and Models of Kinetic Theory (M&MKT), Porto Ercole. Grosseto, Italy (2010) 177–216. | MR | Zbl
The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Commun. Pure Appl. Math. 48 (1995) 235–276. | DOI | MR | Zbl
and ,Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math. 34 (1981) 481–524. | DOI | MR | Zbl
and ,Compressible and incompressible fluids. Commun. Pure Appl. Math. 35 (1982) 629–651. | DOI | MR | Zbl
and ,Convergence of the solutions of the compressible to the solutions of the incompressible Navier–Stokes equations. Adv. Appl. Math. 12 (1991) 187–214. | DOI | MR | Zbl
, and ,Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes. J. Comput. Phys. 69 (1987) 283–324. | DOI | MR | Zbl
, and ,Convergence of a relaxation scheme for hyperbolic systems of conservation laws. Numer. Math. 88 (2001) 121–134. | DOI | MR | Zbl
and ,Finite energy solutions to the isentropic Euler equations with geometric effects. J. Math. Pures Appl. 88 (2007) 389–429. | DOI | MR | Zbl
and ,Hyperbolic conservation laws with relaxation. Commun. Math. Phys. 108 (1987) 153–175. | DOI | MR | Zbl
,N. Masmoudi, Examples of singular limits in hydrodynamics. In Vol. 3 of Handbook of Differential Equations: Evolutionary Equations (2007) 195–275. | MR | Zbl
Asymptotic single and multiple scale expansions in the low Mach number limit. SIAM J. Appl. Math. 60 (1999) 256–271. | DOI | MR | Zbl
,The incompressible limit of the non-isentropic Euler equations. Arch. Ratio. Mech. Anal. 158 (2001) 61–90. | DOI | MR | Zbl
and ,K. Revi Arun, M. Lukcáˇová-Medviďováand C.-D. Munz, A weakly asymptotic preserving low Mach number scheme for the Euler equations of gas dynamics. SIAM J. Sci. Comput. 36 (2014) B989–B1024. | DOI | MR | Zbl
, ,On the dissipation mechanism of upwind-schemes in the low Mach number regime: A comparison between Roe and HLL. J. Comput. Phys. 229 (2010) 221–232. | DOI | MR | Zbl
,A low-Mach number fix for Roe’s approximate Riemann solver. J. Comput. Phys. 230 (2011) 5263–5287. | DOI | MR | Zbl
,The influence of cell geometry on the accuracy of upwind schemes in the low Mach number regime. J. Comput. Phys. 228 (2009) 2918–2933. | DOI | MR | Zbl
and ,J. Schütz and S. Noelle, Flux splitting for stiff equations: A notion on stability. J. Sci. Comput. (2014) 1–19. | MR
A lower bound for the smallest singular value of a matrix. Linear Algebra Appl. 11 (1975) 3–5. | DOI | MR | Zbl
,Shock waves in arbitrary fluids. Commun. Pure Appl. Math. 2 (1949) 103–122. | DOI | MR | Zbl
,H. Zakerzadeh and S. Noelle, A note on the stability of implicit-explicit flux-splittings for stiff systems of hyperbolic conservation laws. IGPM report 449, RWTH Aachen University (2016). | MR
Cité par Sources :