Asymptotic analysis of the RS-IMEX scheme for the shallow water equations in one space dimension
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 3, pp. 893-924.

We introduce and analyse the so-called Reference Solution IMplicit-EXplicit scheme as a flux-splitting method for singularly-perturbed systems of balance laws. RS-IMEX scheme’s bottom-line is to use the Taylor expansion of the flux function and the source term around a reference solution (typically the asymptotic limit or an equilibrium solution) to decompose the flux and the source into stiff and non-stiff parts so that the resulting IMEX scheme is Asymptotic Preserving (AP) w.r.t. the singular parameter tending to zero. We prove the asymptotic consistency, asymptotic stability, solvability and well-balancing of the scheme for the case of the one-dimensional shallow water equations when the singular parameter is the Froude number. We will also study several test cases to illustrate the quality of the computed solutions and to confirm the analysis.

DOI : 10.1051/m2an/2019005
Classification : 35L65, 65M08, 35L81, 65M12
Mots-clés : IMEX scheme, asymptotic preserving, flux splitting, stability analysis
Zakerzadeh, Hamed 1

1 
@article{M2AN_2019__53_3_893_0,
     author = {Zakerzadeh, Hamed},
     title = {Asymptotic analysis of the {RS-IMEX} scheme for the shallow water equations in one space dimension},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {893--924},
     publisher = {EDP-Sciences},
     volume = {53},
     number = {3},
     year = {2019},
     doi = {10.1051/m2an/2019005},
     mrnumber = {3973920},
     zbl = {1450.65098},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2019005/}
}
TY  - JOUR
AU  - Zakerzadeh, Hamed
TI  - Asymptotic analysis of the RS-IMEX scheme for the shallow water equations in one space dimension
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2019
SP  - 893
EP  - 924
VL  - 53
IS  - 3
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2019005/
DO  - 10.1051/m2an/2019005
LA  - en
ID  - M2AN_2019__53_3_893_0
ER  - 
%0 Journal Article
%A Zakerzadeh, Hamed
%T Asymptotic analysis of the RS-IMEX scheme for the shallow water equations in one space dimension
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2019
%P 893-924
%V 53
%N 3
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2019005/
%R 10.1051/m2an/2019005
%G en
%F M2AN_2019__53_3_893_0
Zakerzadeh, Hamed. Asymptotic analysis of the RS-IMEX scheme for the shallow water equations in one space dimension. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 3, pp. 893-924. doi : 10.1051/m2an/2019005. http://www.numdam.org/articles/10.1051/m2an/2019005/

[1] S. Abarbanel, P. Duth and D. Gottlieb, Splitting methods for low Mach number Euler and Navier–Stokes equations. Comput. Fluids 17 (1989) 1–12. | DOI | Zbl

[2] K.R. Arun and S. Noelle, An Asymptotic Preserving Scheme for Low Froude Number Shallow Flows. IGPM Report 352. RWTH Aachen University (2012).

[3] U.M. Ascher, S.J. Ruuth and R.J. Spiteri, Implicit-explicit Runge–Kutta methods for time-dependent partial differential equations. Appl. Numer. Math. 25 (1997) 151–167. | DOI | MR | Zbl

[4] I. Bendixson, Sur les racines d’une équation fondamentale. Acta Math. 25 (1902) 359–365. | DOI | JFM | MR

[5] D.S. Bernstein, Matrix Mathematics: Theory, Facts, and Formulas. Princeton University Press, Princeton, NJ (2009). | DOI | MR | Zbl

[6] G. Bispen, IMEX finite volume methods for the shallow water equationsPh.D. thesis, Johannes Gutenberg-Universität Mainz (2015).

[7] G. Bispen, K.R. Arun, M. Lukáčová-Medvid’Ová and S. Noelle, IMEX large time step finite volume methods for low Froude number shallow water flows. Commun. Comput. Phys. 16 (2014) 307–347. | DOI | MR | Zbl

[8] G. Bispen, M. Lukáčová-Medvid’Ová and L. Yelash, Asymptotic preserving IMEX finite volume schemes for low mach number Euler equations with gravitation. J. Comput. Phys. 335 (2017) 222–248. | DOI | MR | Zbl

[9] S. Boscarino and G. Russo, On a class of uniformly accurate IMEX Runge–Kutta schemes and applications to hyperbolic systems with relaxation. SIAM J. Sci. Comput. 31 (2009) 1926–1945. | DOI | MR | Zbl

[10] F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws: And Well-Balanced Schemes for Sources. Springer Science Business Media, Berlin (2004) | DOI | MR | Zbl

[11] F. Bouchut and M. Westdickenberg, Gravity driven shallow water models for arbitrary topography. Commun. Math. Sci. 2 (2004) 359–389. | DOI | MR | Zbl

[12] D. Bresch, R. Klein and C. Lucas, Multiscale analyses for the shallow water equations. In: Computational Science and High Performance Computing IV. Springer, Berlin (2011) 149–164. | MR

[13] R.E. Caflisch, S. Jin, G. Russo, Uniformly accurate schemes for hyperbolic systems with relaxation. SIAM J. Numer. Anal. 34 (1997) 246–281. | DOI | MR | Zbl

[14] F. Cordier, P. Degond and A. Kumbaro, An asymptotic-preserving all-speed scheme for the Euler and Navier–Stokes equations. J. Comput. Phys. 231 (2012) 5685–5704. | DOI | MR | Zbl

[15] R. Danchin, Low Mach number limit for viscous compressible flows. ESAIM: M2AN 39 (2005) 459–475. | DOI | Numdam | MR | Zbl

[16] P. Degond and M. Tang, All speed scheme for the low Mach number limit of the isentropic Euler equation. Commun. Comput. Phys. 10 (2011) 1–31. | DOI | MR | Zbl

[17] P. Degond, J.-G. Liu and M.-H. Vignal, 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

[18] S. Dellacherie, 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

[19] S. Dellacherie, P. Omnes and F. Rieper, 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

[20] G. Dimarco, R. Loubère and M.-H. Vignal, Study of a new asymptotic preserving scheme for the Euler system in the low Mach number limit. SIAM J. Sci. Comput. 39 (2017) A2099–A2128. | DOI | MR | Zbl

[21] D. Drikakis and W. Rider, High-resolution Methods for Incompressible and Low-speed Flows, Springer Science & Business Media, Berlin (2006). | MR

[22] B. Fedele and C. Negulescu, Numerical study of an anisotropic Vlasov equation arising in plasma physics. Preprint arXiv:1610.01592 (2016) . | MR | Zbl

[23] E. Feireisl and A. Novotnỳ, Singular Limits in Thermodynamics of Viscous Fluids, Springer Science & Business Media, Berlin (2009). | DOI | MR | Zbl

[24] E. Feireisl, M. Lukáčová-Medvid’Ová, V. Nečasová, A. Novotnỳ and B. She, Asymptotic preserving error estimates for numerical solutions of compressible Navier–Stokes equations in the low Mach number regime. Multiscale Model. Simul. 16 (2018) 150–183. | DOI | MR | Zbl

[25] M. Feistauer, V. Dolejší, V. Kučera, On the discontinuous Galerkin method for the simulation of compressible flow with wide range of Mach numbers. Comput. Visual. Sci. 10 (2007) 17–27. | DOI | MR

[26] F. Filbet and S. Jin, A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources. J. Comput. Phys. 229 (2010) 7625–7648. | DOI | MR | Zbl

[27] T. Gallouët, R. Herbin, D. Maltese and A. Novotnỳ, Implicit MAC scheme for compressible Navier–Stokes equations: low Mach asymptotic error estimates. Preprint hal-01462822 (2017).

[28] J. Giesselmann, Low Mach asymptotic-preserving scheme for the Euler–Korteweg model. IMA J. Numer. Anal. 35 (2015) 802–833. | DOI | MR | Zbl

[29] G.H. Golub and C.F. Van Loan, Matrix Computations. JHU Press, Baltimore, MD 3 (2012). | MR | Zbl

[30] R.M. Gray, Toeplitz and Circulant Matrices: A Review, Now Publishers Inc., Boston, MA (2006). | Zbl

[31] H. Guillard and A. Murrone, On the behavior of upwind schemes in the low Mach number limit: II. Godunov type schemes. Comput. Fluids 33 (2004) 655–675. | DOI | Zbl

[32] H. Guillard and C. Viozat, On the behaviour of upwind schemes in the low Mach number limit. Comput. Fluids 28 (1999) 63–86. | DOI | MR | Zbl

[33] J. Haack, S. Jin and J.-G. Liu, An all-speed asymptotic-preserving method for the isentropic Euler and Navier–Stokes equations. Commun. Comput. Phys. 12 (2012) 955–980. | DOI | MR | Zbl

[34] E. Hairer and G. Wanner, Solving ordinary differential equations. II, Stiff and differential-algebraic problems, 2nd edition. In: Vol. 14 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin (1996). | MR | Zbl

[35] A. Hiltebrand and S. Mishra, Efficient computation of all speed flows using an entropy stable shock-capturing space-time discontinuous Galerkin method, Non-Linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis. European Mathematical Society (2018) 287–318. | DOI | MR | Zbl

[36] M.A. Hirsch, Sur les racines d’une équation fondamentale. Acta Math. 25 (1902) 367–370. | JFM | MR

[37] R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, New York, NY, USA (1986). | MR | Zbl

[38] R.A. Horn and C.R. Johnson, Topics in Matrix Analysis, Cambridge UP, New York, NY (1991). | DOI | MR

[39] J. Hu, S. Jin and Q. Li, Asymptotic-preserving schemes for multiscale hyperbolic and kinetic equations. In: Handbook of Numerical Analysis (2016). | MR | Zbl

[40] S. Jin, Runge–Kutta methods for hyperbolic conservation laws with stiff relaxation terms. J. Comput. Phys. 122 (1995) 51–67. | DOI | MR | Zbl

[41] S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21 (1999) 441–454. | DOI | MR | Zbl

[42] S. Jin, Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review. . In: Lecture Notes for Summer School on “Methods and Models of Kinetic Theory” (M&MKT), Porto Ercole (Grosseto, Italy) (2010) 177–216. | MR | Zbl

[43] K. Kaiser, J. Schütz, R. Schöbel and S. Noelle, A new stable splitting for the isentropic Euler equations. J. Sci. Comput. 70 (2017) 1390–1407. | DOI | MR | Zbl

[44] S. Klainerman and A. Majda, 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

[45] S. Klainerman and A. Majda, Compressible and incompressible fluids. Commun. Pure Appl. Math. 35 (1982) 629–651. | DOI | MR | Zbl

[46] A. Klar, A numerical method for nonstationary transport equations in diffusive regimes. Transp. Theory Stat. Phys. 27 (1998) 653–666. | DOI | MR | Zbl

[47] R. Klein, Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics I: one-dimensional flow. J. Comput. Phys. 121 (1995) 213–237. | DOI | MR | Zbl

[48] R. Klein, N. Botta, T. Schneider, C.-D. Munz, S. Roller, A. Meister, L. Hoffmann and T. Sonar, Asymptotic adaptive methods for multi-scale problems in fluid mechanics. In: Practical Asymptotics. Springer, Berlin (2001) 261–343. | DOI | MR | Zbl

[49] E.W. Larsen, J.E. Morel and W.F. Miller, Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes. J. Comput. Phys. 69 (1987) 283–324. | DOI | MR | Zbl

[50] N. Masmoudi, Examples of singular limits in hydrodynamics.In: Vol. 3 of Handbook of Differential Equations: Evolutionary Equations (2007) 195–275. | MR | Zbl

[51] H. Nessyahu and E. Tadmor, The convergence rate of approximate solutions for nonlinear scalar conservation laws. SIAM J. Numer. Anal. 29 (1992) 1505–1519. | DOI | MR | Zbl

[52] S. Noelle, G. Bispen, K.R. Arun, M. Lukáčová-Medvid’Ová and 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

[53] L. Pareschi and G. Russo, Implicit-explicit Runge–Kutta schemes and applications to hyperbolic systems with relaxation. J. Sci. Comput. 25 (2005) 129–155. | MR | Zbl

[54] M. Restelli, , Semi-Lagrangian and semi-implicit discontinuous Galerkin methods for atmospheric modeling applications.Ph.D. thesis, Politecnico di Milano (2007).

[55] R.D. Richtmyer and K.W. Morton, Difference Methods for Initial-value Problems, Interscience Publishers John Wiley & Sons, Inc, Academia Publishing House of the Czechoslovak Acad (1967). | MR | Zbl

[56] F. Rieper, 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

[57] F. Rieper, A low-Mach number fix for Roe’s approximate Riemann solver. J. Comput. Phys. 230 (2011) 5263–5287. | DOI | MR | Zbl

[58] F. Rieper and G. Bader, 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

[59] H.H. Rosenbrock, Some general implicit processes for the numerical solution of differential equations. Comput. J. 5 (1963) 329–330. | DOI | MR | Zbl

[60] S. Schochet, The mathematical theory of low Mach number flows. ESAIM: M2AN 39 (2005) 441–458. | DOI | Numdam | MR | Zbl

[61] J. Schütz and K. Kaiser, A new stable splitting for singularly perturbed ODEs. Appl. Numer. Math. 107 (2016) 18–33. | DOI | MR | Zbl

[62] J. Schütz and S. Noelle, Flux splitting for stiff equations: a notion on stability. J. Sci. Comput. 64 (2015) 522–540. | DOI | MR | Zbl

[63] J.R. Silvester, Determinants of block matrices. Math. Gazette 84 (2000) 460–467. | DOI

[64] L.N. Trefethen, Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations, Cornell University (1996).

[65] R.F. Warming and B.J. Hyett, The modified equation approach to the stability and accuracy analysis of finite-difference methods. J. Comput. Phys. 14 (1974) 159–179. | DOI | MR | Zbl

[66] D. Willett and J.S.W. Wong, On the discrete analogues of some generalizations of Gronwall’s inequality. Monatsh. Math. 69 (1965) 362–367. | DOI | MR | Zbl

[67] H. Zakerzadeh, Asymptotic preserving finite volume schemes for the singularly-perturbed shallow water equations with source terms. PhD thesis, RWTH Aachen University (2017)

[68] H. Zakerzadeh, On the Mach-uniformity of the Lagrange-projection scheme. ESAIM: M2AN 51 (2017) 1343–1366. | Numdam | MR | Zbl

[69] H. Zakerzadeh, The RS-IMEX Scheme for the Rotating Shallow Water Equations with the Coriolis Force. Springer International Publishing, Cham (2017) 199–207. | MR

[70] H. Zakerzadeh, Asymptotic Consistency of the RS-IMEX Scheme for the low-Froude Shallow Water Equations: Analysis and Numerics. Springer International Publishing, Cham (2018) 665–675. | MR

[71] H. Zakerzadeh and S. Noelle, A note on the stability of implicit-explicit flux-splittings for stiff systems of hyperbolic conservation laws. Commun. Math. Sci. 16 (2018) 1–15. | DOI | MR | Zbl

[72] M. Zakerzadeh and G. May, On the convergence of a shock capturing discontinuous Galerkin method for nonlinear hyperbolic systems of conservation laws. SIAM J. Numer. Anal. 54 (2016) 874–898. | DOI | MR | Zbl

Cité par Sources :