Conservativity and weak consistency of a class of staggered finite volume methods for the Euler equations
HTML articles powered by AMS MathViewer
- by R. Herbin, J.-C. Latché, S. Minjeaud and N. Therme;
- Math. Comp. 90 (2021), 1155-1177
- DOI: https://doi.org/10.1090/mcom/3575
- Published electronically: December 29, 2020
- HTML | PDF | Request permission
Abstract:
We address here a class of staggered schemes for the compressible Euler equations; this scheme was introduced in recent papers and possesses the following features: upwinding is performed with respect to the material velocity only and the internal energy balance is solved, with a correction term designed on consistency arguments. These schemes have been shown in previous works to preserve the convex of admissible states and have been extensively tested numerically. The aim of the present paper is twofold: we derive a local total energy equation satisfied by the solutions, so that the schemes are in fact conservative, and we prove that they are consistent in the Lax-Wendroff sense.References
- G. Ansanay-Alex, F. Babik, J. C. Latché, and D. Vola, An $L^2$-stable approximation of the Navier-Stokes convection operator for low-order non-conforming finite elements, Internat. J. Numer. Methods Fluids 66 (2011), no. 5, 555–580. MR 2839213, DOI 10.1002/fld.2270
- David J. Benson, Computational methods in Lagrangian and Eulerian hydrocodes, Comput. Methods Appl. Mech. Engrg. 99 (1992), no. 2-3, 235–394. MR 1186728, DOI 10.1016/0045-7825(92)90042-I
- Hester Bijl and Pieter Wesseling, A unified method for computing incompressible and compressible flows in boundary-fitted coordinates, J. Comput. Phys. 141 (1998), no. 2, 153–173. MR 1619651, DOI 10.1006/jcph.1998.5914
- CALIF$^3$S, A software components library for the computation of reactive turbulent flows, https://gforge.irsn.fr/gf/project/isis.
- E. J. Caramana, D. E. Burton, M. J. Shashkov, and P. P. Whalen, The construction of compatible hydrodynamics algorithms utilizing conservation of total energy, J. Comput. Phys. 146 (1998), no. 1, 227–262. MR 1650480, DOI 10.1006/jcph.1998.6029
- Gautier Dakin, Bruno Després, and Stéphane Jaouen, High-order staggered schemes for compressible hydrodynamics. Weak consistency and numerical validation, J. Comput. Phys. 376 (2019), 339–364. MR 3875525, DOI 10.1016/j.jcp.2018.09.046
- Robert Eymard and Thierry Gallouët, $H$-convergence and numerical schemes for elliptic problems, SIAM J. Numer. Anal. 41 (2003), no. 2, 539–562. MR 2004187, DOI 10.1137/S0036142901397083
- T. Gallouët, R. Herbin, and J.-C. Latché, On the weak consistency of finite volumes schemes for conservation laws on general meshes, SeMA J. 76 (2019), no. 4, 581–594. MR 4025828, DOI 10.1007/s40324-019-00194-x
- T. Gallouët, R. Herbin, J.-C. Latché, and N. Therme, Consistent internal energy based schemes for the compressible Euler equations, Numerical Simulation in Physics and Engineering: Trends and Applications, Lecture Notes of the XVIII Jacques-Louis Lions Spanish-French School, Springer, SEMA/SIMAI Series, 2020.
- Laura Gastaldo, Raphaèle Herbin, Jean-Claude Latché, and Nicolas Therme, A MUSCL-type segregated–explicit staggered scheme for the Euler equations, Comput. & Fluids 175 (2018), 91–110. MR 3864518, DOI 10.1016/j.compfluid.2018.06.013
- Dionysis Grapsas, Raphaèle Herbin, Walid Kheriji, and Jean-Claude Latché, An unconditionally stable staggered pressure correction scheme for the compressible Navier-Stokes equations, SMAI J. Comput. Math. 2 (2016), 51–97. MR 3633545, DOI 10.5802/smai-jcm.9
- F.H. Harlow and A.A. Amsden, A numerical fluid dynamics calculation method for all flow speeds, Journal of Computational Physics 8 (1971), 197–213.
- Francis H. Harlow and J. Eddie Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluids 8 (1965), no. 12, 2182–2189. MR 3155392, DOI 10.1063/1.1761178
- R. Herbin, W. Kheriji, and J.-C. Latché, On some implicit and semi-implicit staggered schemes for the shallow water and Euler equations, ESAIM Math. Model. Numer. Anal. 48 (2014), no. 6, 1807–1857. MR 3342164, DOI 10.1051/m2an/2014021
- Raphaèle Herbin, Jean-Claude Latché, and Trung Tan Nguyen, Consistent segregated staggered schemes with explicit steps for the isentropic and full Euler equations, ESAIM Math. Model. Numer. Anal. 52 (2018), no. 3, 893–944. MR 3865553, DOI 10.1051/m2an/2017055
- R. Herbin, J.-C. Latché, and K. Saleh, Low Mach number limit of some staggered schemes for compressible barotropic flows, under revision (2020), https://arxiv.org/abs/1803.09568.
- Raphaèle Herbin, Jean-Claude Latché, and Chady Zaza, A cell-centred pressure-correction scheme for the compressible Euler equations, IMA J. Numer. Anal. 40 (2020), no. 3, 1792–1837. MR 4122491, DOI 10.1093/imanum/drz024
- R. I. Issa, Solution of the implicitly discretised fluid flow equations by operator-splitting, J. Comput. Phys. 62 (1986), no. 1, 40–65. MR 825890, DOI 10.1016/0021-9991(86)90099-9
- K.C. Karki and S.V. Patankar, Pressure based calculation procedure for viscous flows at all speeds in arbitrary configurations, AIAA Journal 27 (1989), 1167–1174.
- Nipun Kwatra, Jonathan Su, Jón T. Grétarsson, and Ronald Fedkiw, A method for avoiding the acoustic time step restriction in compressible flow, J. Comput. Phys. 228 (2009), no. 11, 4146–4161. MR 2524514, DOI 10.1016/j.jcp.2009.02.027
- J.-C. Latché, B. Piar, and K. Saleh, A discrete kinetic energy preserving convection operator for variable density flows on locally refined staggered meshes., in preparation (2019).
- J. Llobell, Schémas volumes finis à mailles décalées pour la dynamique des gaz, Ph.D. thesis, Université Côte d’Azur, 2018.
- J. J. McGuirk and G. J. Page, Shock capturing using a pressure-correction method, AIAA Journal 28 (1990), 1751–1757.
- R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element, Numer. Methods Partial Differential Equations 8 (1992), no. 2, 97–111. MR 1148797, DOI 10.1002/num.1690080202
- R. W. C. P. Verstappen and A. E. P. Veldman, Symmetry-preserving discretization of turbulent flow, J. Comput. Phys. 187 (2003), no. 1, 343–368. MR 1977790, DOI 10.1016/S0021-9991(03)00126-8
- J. Von Neumann and R. D. Richtmyer, A method for the numerical calculation of hydrodynamic shocks, J. Appl. Phys. 21 (1950), 232–237. MR 37613
- Clifton Wall, Charles D. Pierce, and Parviz Moin, A semi-implicit method for resolution of acoustic waves in low Mach number flows, J. Comput. Phys. 181 (2002), no. 2, 545–563. MR 1927401, DOI 10.1006/jcph.2002.7141
- Pieter Wesseling, Principles of computational fluid dynamics, Springer Series in Computational Mathematics, vol. 29, Springer-Verlag, Berlin, 2001. MR 1796357, DOI 10.1007/978-3-642-05146-3
- S.Y. Yoon and T. Yabe, The unified simulation for incompressible and compressible flow by the predictor-corrector scheme based on the CIP method, Computer Physics Communications 119 (1999), 149–158.
Bibliographic Information
- R. Herbin
- Affiliation: I2M, CNRS and Université d’Aix-Marseille
- MR Author ID: 244425
- ORCID: 0000-0003-0937-1900
- Email: raphaele.herbin@univ-amu.fr
- J.-C. Latché
- Affiliation: Institut de Radioprotection et de Sûreté Nucléaire (IRSN), PSN-RES/SA2I, Cadarache, St-Paul-lez-Durance, 13115, France
- MR Author ID: 715367
- Email: jean-claude.latche@irsn.fr
- S. Minjeaud
- Affiliation: Laboratoire Jean Dieudonné, Université de Nice-Sophia Antipolis
- MR Author ID: 889818
- Email: sebastian.minjeaud@unice.fr
- N. Therme
- Affiliation: CEA/CESTA 33116, Le Barp, France
- MR Author ID: 1088061
- Email: nicolas.therme@cea.fr
- Received by editor(s): December 8, 2019
- Received by editor(s) in revised form: June 6, 2020
- Published electronically: December 29, 2020
- © Copyright 2020 American Mathematical Society
- Journal: Math. Comp. 90 (2021), 1155-1177
- MSC (2010): Primary 65M08
- DOI: https://doi.org/10.1090/mcom/3575
- MathSciNet review: 4232220