An unconditionally stable staggered pressure correction scheme for the compressible Navier-Stokes equations

Grapsas, Dionysis; Herbin, Raphaèle; Kheriji, Walid; Latché, Jean-Claude

The SMAI Journal of computational mathematics, Tome 2 (2016), pp. 51-97.

In this paper we present a pressure correction scheme for the compressible Navier-Stokes equations. The space discretization is staggered, using either the Marker-And-Cell (MAC) scheme for structured grids, or a nonconforming low-order finite element approximation for general quandrangular, hexahedral or simplicial meshes. For the energy balance equation, the scheme uses a discrete form of the conservation of the internal energy, which ensures that this latter variable remains positive; this relation includes a numerical corrective term, to allow the scheme to compute correct shock solutions in the Euler limit. The scheme is shown to have at least one solution, and to preserve the stability properties of the continuous problem, irrespectively of the space and time steps. In addition, it naturally boils down to a usual projection scheme in the limit of vanishing Mach numbers. Numerical tests confirm its potentialities, both in the viscous incompressible and Euler limits.

Classification : 65M08, 76N15, 65M12, 76N19
Mots-clés : Compressible Navier-Stokes equations, pressure correction schemes, finite volumes, MAC scheme, finite elements.

[1] Bijl, H.; Wesseling, P. A Unified Method for Computing Incompressible and Compressible Flows in Boundary-Fitted Coordinates, Journal of Computational Physics, Volume 141 (1998), pp. 153-173 | DOI | MR | Zbl

[2] Boyer, F.; Dardalhon, F.; Lapuerta, C.; Latché, J.-C. Stability of a Crank-Nicolson pressure correction scheme based on staggered discretizations, International Journal for Numerical Methods in Fluids, Volume 74 (2014), pp. 34-58 | DOI | MR

[3] CALIF3S A software components library for the computation of reactive turbulent flows (https://gforge.irsn.fr/gf/project/calif3s)

[4] Casulli, V.; Greenspan, D. Pressure Method for the Numerical Solution of Transient, Compressible Fluid Flows, International Journal for Numerical Methods in Fluids, Volume 4 (1984), pp. 1001-1012 | DOI | Zbl

[5] Ciarlet, P. G. Handbook of Numerical Analysis Volume II : Finite Elements Methods – Basic Error Estimates for Elliptic Problems, Handbook of Numerical Analysis, Volume II (Ciarlet, P.; Lions, J.L., eds.), North Holland, 1991, pp. 17-351 | DOI

[6] Clain, S.; Diot, S.; Loubère, R. A high-order finite volume method for systems of conservation laws – Multi-dimensional Optimal Order Detection (MOOD), Journal of Computational Physics, Volume 230 (2011), pp. 4028-4050 | DOI | MR | Zbl

[7] Cockburn, B.; Shu, C.-W. The Runge-Kutta Discontinuous Galerkin method for conservation laws V – Multidimensional systems, Journal of Computational Physics, Volume 141 (1998), pp. 199-224 | DOI | MR | Zbl

[8] Colella, P.; Pao, K. A projection method for low speed flows, Journal of Computational Physics, Volume 149 (1999), pp. 245-269 | DOI | MR | Zbl

[9] Crouzeix, M.; Raviart, P.A. Conforming and nonconforming finite element methods for solving the stationary Stokes equations, RAIRO Série Rouge, Volume 7 (1973), pp. 33-75 | Numdam | MR

[10] Dardalhon, F.; Latché, J.-C.; Minjeaud, S. Analysis of a projection method for low-order non-conforming finite elements, IMA Journal of Numerical Analysis, Volume 33 (2013), pp. 295-317 | DOI | Zbl

[11] Drábek, P.; Milota, J. Methods of Nonlinear Analysis, Birkhäuser Advanced Texts, Basel Lehrbücher, 2007

[12] Eymard, R.; Gallouët, T.; Herbin, R. Finite Volume Methods, Handbook of Numerical Analysis, Volume VII (Ciarlet, P.; Lions, J.L., eds.), North Holland, 2000, pp. 713-1020 | DOI

[13] Gastaldo, L.; Herbin, R.; Kheriji, W.; Lapuerta, C.; Latché, J.-C. Staggered discretizations, pressure correction schemes and all speed barotropic flows, Finite Volumes for Complex Applications VI - Problems & Perspectives - Prague, Czech Republic, Volume 2, Springer (2011), pp. 39-56 | Zbl

[14] Gastaldo, L.; Herbin, R.; Latché, J.-C. A discretization of the phase mass balance in fractional step algorithms for the drift-flux model, IMA Journal of Numerical Analysis, Volume 31 (2011), pp. 116-146 | DOI | MR | Zbl

[15] Guermond, J.-L.; Quartapelle, L. A Projection FEM for Variable Density Incompressible Flows, Journal of Computational Physics, Volume 165 (2000), pp. 167-188 | DOI | MR | Zbl

[16] Guermond, J.L.; Pasquetti, R.; Popov, B. Entropy viscosity method for nonlinear conservation laws, Journal of Computational Physics, Volume 230 (2011), pp. 4248-4267 | DOI | MR | Zbl

[17] Harlow, F.H.; Amsden, A.A. Numerical Calculation of Almost Incompressible Flow, Journal of Computational Physics, Volume 3 (1968), pp. 80-93 | DOI

[18] Harlow, F.H.; Amsden, A.A. A numerical fluid dynamics calculation method for all flow speeds, Journal of Computational Physics, Volume 8 (1971), pp. 197-213 | DOI | Zbl

[19] Harlow, F.H.; Welsh, J.E. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Physics of Fluids, Volume 8 (1965), pp. 2182-2189 | DOI | MR

[20] Herbin, R.; Kheriji, W.; Latché, J.-C. On some implicit and semi-implicit staggered schemes for the shallow water and Euler equations., Mathematical Modelling and Numerical Analysis, Volume 48 (2014), pp. 1807-1857 | DOI | Numdam | MR | Zbl

[21] Herbin, R.; Latché, J.-C. Kinetic energy control in the MAC discretisation of the compressible Navier-Stokes equations, International Journal of Finite Volumes, Volume 7 (2010)

[22] Herbin, R.; Latché, J.-C.; Nguyen, T.T. On some consistent explicit staggered schemes for the shallow water and Euler equations., under revision (2016)

[23] Herbin, R.; Latché, J.-C.; Zaza, C. A cell-centered pressure correction scheme for the compressible Euler equations, under revision (2016)

[24] Issa, R.I. Solution of the Implicitly Discretised Fluid Flow Equations by Operator Splitting, Journal of Computational Physics, Volume 62 (1985), pp. 40-65 | DOI | MR

[25] Issa, R.I.; Gosman, A.D.; Watkins, A.P. The Computation of Compressible and Incompressible Recirculating Flows by a Non-iterative Implicit Scheme, Journal of Computational Physics, Volume 62 (1986), pp. 66-82 | DOI

[26] Karki, K.C.; Patankar, S.V. Pressure Based Calculation Procedure for Viscous Flows at All Speeds in Arbitrary Configurations, AIAA Journal, Volume 27 (1989), pp. 1167-1174 | DOI

[27] Kheriji, W. Méthodes de correction de pression pour les équations de Navier-Stokes compressibles, Université de Provence (2011) (Thèse de Mathématiques Appliquées)

[28] Kheriji, W.; Herbin, R.; Latché, J.-C. Pressure correction staggered schemes for barotropic monophasic and two-phase flows, Computers & Fluids, Volume 88 (2013), pp. 524-542 | DOI | Zbl

[29] Kurganov, A.; Tadmor, E. Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers, Numerical Methods for Partial Differential Equations, Volume 18 (2002), pp. 584-608 | DOI | MR | Zbl

[30] Kwatra, N.; Su, J.; Grétarsson, J.T.; Fedkiw, R. A method for avoiding the acoustic time step restriction in compressible flow, Journal of Computational Physics, Volume 228 (2009), pp. 4146-4161 | DOI | MR | Zbl

[31] Larrouturou, B. How to Preserve the Mass Fractions Positivity when Computing Compressible Multi-Component Flows, Journal of Computational Physics, Volume 95 (1991), pp. 59-84 | DOI | MR | Zbl

[32] Lax, P.; Liu, X.-D. Solution of two-dimensional Riemann problems of gas dynamics by positive schemes, SIAM Journal on Scientific Computing, Volume 19 (1998), pp. 319-340 | DOI | MR

[33] Liu, C.; Walkington, N.J. Convergence of numerical approximations of the incompressible Navier-Stokes equations with variable density and viscosity, SIAM Journal on Numerical Analysis, Volume 45 (2007), pp. 1287-1304 | DOI | MR

[34] McGuirk, J.J.; Page, G.J. Shock capturing using a pressure-correction method, AIAA Journal, Volume 28 (1990), pp. 1751-1757 | DOI

[35] Minjeaud, S. An unconditionally stable uncoupled scheme for a triphasic Cahn-Hilliard/Navier-Stokes model, Numerical Methods for Partial Differential Equations, Volume 29 (2013), pp. 584-618 | DOI | MR | Zbl

[36] Nerinckx, K.; Vierendeels, J.; Dick, E. Mach-uniformity through the coupled pressure and temperature correction algorithm, Journal of Computational Physics, Volume 206 (2005), pp. 597-623 | DOI | Zbl

[37] Nerinckx, K.; Vierendeels, J.; Dick, E. A Mach-uniform algorithm: coupled versus segregated approach, Journal of Computational Physics, Volume 224 (2007), pp. 314-331 | DOI | MR | Zbl

[38] Piar, L.; Babik, F.; Herbin, R.; Latché, J.-C. A formally second order cell centered scheme for convection-diffusion equations on general grids, International Journal for Numerical Methods in Fluids, Volume 71 (2013), pp. 873-890 | DOI | Zbl

[39] Rannacher, R.; Turek, S. Simple nonconforming quadrilateral Stokes element, Numerical Methods for Partial Differential Equations, Volume 8 (1992), pp. 97-111 | DOI | MR | Zbl

[40] Schäfer, M.; Turek, S. Benchmark Computations of Laminar Flow Around a Cylinder, Notes on Numerical Fluid Mechanics, 52, Vieweg, 1996, pp. 547-566 | Zbl

[41] Schulz-Rinne, C.W.; Collins, J.P.; Glaz, H.M. Numerical solution of the Riemann problem for two-dimensional gas dynamics, SIAM Journal on Scientific Computing, Volume 14 (1993), pp. 1394-1414 | DOI | MR

[42] Van der Heul, D.R.; Vuik, C.; Wesseling, P. Stability Analysis of Segregated Solution Methods for Compressible Flow, Applied Numerical Mathematics, Volume 38 (2001), pp. 257-274 | DOI | MR | Zbl

[43] Van der Heul, D.R.; Vuik, C.; Wesseling, P. A conservative pressure-correction method for flow at all speeds, Computers & Fluids, Volume 32 (2003), pp. 1113-1132 | DOI | MR | Zbl

[44] Van Dormaal, J.P.; Raithby, G.D.; McDonald, B.H. The Segregated Approach to Predicting Viscous Compressible Fluid Flows, Transactions of the ASME, Volume 109 (1987), pp. 268-277

[45] Vidović, D.; Segal, A.; Wesseling, P. A superlinearly convergent Mach-uniform finite volume method for the Euler equations on staggered unstructured grids, Journal of Computational Physics, Volume 217 (2006), pp. 277-294 | DOI | MR | Zbl

[46] Wall, C.; Pierce, C.D.; Moin, P. A Semi-implicit Method for Resolution of Acoustic Waves in Low Mach Number Flows, Journal of Computational Physics, Volume 181 (2002), pp. 545-563 | DOI | MR | Zbl

[47] Wenneker, I.; Segal, A.; Wesseling, P. A Mach-Uniform Unstructured Staggered Grid Method, International Journal for Numerical Methods in Fluids, Volume 40 (2002), pp. 1209-1235 | DOI | MR | Zbl

[48] Woodward, P.; Colella, P. The numerical simulation of two-dimensional fluid flow with strong shocks, Journal of Computational Physics, Volume 54 (1984), pp. 115-173 | DOI | MR | Zbl

[49] Xu, Z.; Liu, Y.; Du, H.; Lin, G.; Shu, C.-W. Point-wise hierarchical reconstruction for discontinuous Galerkin and finite volume methods for solving conservation laws, Journal of Computational Physics, Volume 230 (2011), pp. 6843-6865 | DOI | MR | Zbl

[50] Yoon, S.Y.; Yabe, T. The unified simulation for incompressible and compressible flow by the predictor-corrector scheme based on the CIP method, Computer Physics Communications, Volume 119 (1999), pp. 149-158 | DOI | Zbl