no. 2
Numerical analysis
High-order accurate Lagrange-remap hydrodynamic schemes on staggered Cartesian grids
[Schémas hydrodynamiques d'ordre très élevé Lagrange-projection sur grilles cartésiennes décalées]

Présenté par : the Editorial Board

Dakin, Gautier1 ; Jourdren, Hervé1
1 CEA, DAM, DIF, 91297 Arpajon, France
Comptes Rendus. Mathématique, Tome 354 (2016) no. 2, pp. 211-217.

Nous considérons une classe de schémas sur maillage décalé pour résoudre les équations d'Euler 1D. Les schémas proposés, formulés en énergie interne, sont d'ordre élevé en espace comme en temps, utilisables quelle que soit l'équation d'état. En ajoutant une discrétisation de l'équation de l'énergie cinétique, une procédure de synchronisation de l'énergie cinétique est introduite, préservant globalement l'énergie totale et permettant la capture correcte des chocs. Une extension nD sur grille cartésienne décalée de type C avec splitting directionnel d'ordre élevé est proposée. Des résultats numériques sont présentés jusqu'à l'ordre 8.

We consider a class of staggered grid schemes for solving the 1D Euler equations in internal energy formulation. The proposed schemes are applicable to arbitrary equations of state and high-order accurate in both space and time on smooth flows. Adding a discretization of the kinetic energy equation, a high-order kinetic energy synchronization procedure is introduced, preserving globally total energy and enabling proper shock capturing. Extension to nD Cartesian grids is done via C-type staggering and high-order dimensional splitting. Numerical results are provided up to 8th-order accuracy.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.11.008
Dakin, Gautier 1 ; Jourdren, Hervé 1

1 CEA, DAM, DIF, 91297 Arpajon, France 
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Dakin, Gautier; Jourdren, Hervé. High-order accurate Lagrange-remap hydrodynamic schemes on staggered Cartesian grids. Comptes Rendus. Mathématique, Tome 354 (2016) no. 2, pp. 211-217. doi : 10.1016/j.crma.2015.11.008. http://www.numdam.org/articles/10.1016/j.crma.2015.11.008/

[1] Cook, A.W.; Cabot, W.H. Hyperviscosity for shock-turbulence interactions, J. Comput. Phys., Volume 203 (2005), pp. 379-385

[2] DeBar, R.B. Fundamentals of the KRAKEN code, 1974 (Lawrence Livermore Laboratory Report, Technical Report UCIR-760)

[3] Del Pino, S.; Desprès, B.; Havé, P.; Jourdren, H.; Piserchia, P.F. 3D finite volume simulation of acoustic waves in the Earth atmosphere, Comput. Fluids, Volume 38 (2009) no. 4, pp. 765-777

[4] Duboc, F.; Enaux, C.; Jaouen, S.; Jourdren, H.; Wolff, M. High-order dimensionally split Lagrange-remap schemes for compressible hydrodynamics, C. R. Acad. Sci. Paris, Ser. I, Volume 348 (2010), pp. 105-110

[5] Heuzé, O.; Jaouen, S.; Jourdren, H. Dissipative issue of high-order shock capturing schemes with non-convex equations of state, J. Comput. Phys., Volume 228 (2009), pp. 833-860

[6] Jourdren, H. (LNCSE), Volume vol. 41, Springer (2005), pp. 283-294

[7] Noh, W.F. Numerical methods in hydrodynamic calculations, 1976 (Lawrence Livermore Laboratory Report, Technical Report UCRL-52112)

[8] Popov, Y.P.; Samarskii, A.A. Completely conservative difference schemes, Zh. Vychisl. Mat. Mat. Fiz., Volume 9 (1969) no. 4, pp. 953-958

[9] Richtmyer, R.D. Proposed numerical method for calculation of shocks, 1948 (Los Alamos Report, 671)

[10] Shu, C.-W.; Osher, S. Efficient implementation of essentially non-oscillatory shock-capturing schemes, II, J. Comput. Phys., Volume 83 (1989), pp. 32-78

[11] Sutcliffe, W.G. BBC hydrodynamics, 1974 (Lawrence Livermore Laboratory Report, Technical Report UCID-17013)

[12] Trulio, J.G.; Trigger, K.R. Numerical solution of the one dimensional Lagrangian hydrodynamics equations, 1961 (Lawrence Radiation Laboratory Report, Technical Report UCRL-6267)

[13] Verner, J.H. Jim Verner's Refuge for Runge–Kutta pairs, 2013 http://people.math.sfu.ca/~jverner/

[14] von Neumann, J.; Richtmyer, R.D. A method for numerical calculation of hydrodynamic shocks, J. Appl. Phys., Volume 21 (1950), pp. 232-237

[15] Woodward, P.; Colella, P. The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., Volume 54 (1984), pp. 115-173

[16] Yoshida, H. Construction of higher order symplectic integrators, Phys. Lett. A, Volume 150 (1990), pp. 262-267

[17] Youngs, D.L. The Lagrangian remap method, Implicit Large Eddy Simulation: Computing Turbulent Flow Dynamics, Cambridge University Press, Cambridge, UK, 2007

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