Positivity-preserving dual time stepping schemes for gas dynamics.

*(English)*Zbl 1391.76433Summary: A new approach at discretizing the temporal derivative of the Euler equations is presented which can be used with dual time stepping. The temporal discretization stencil is derived along the lines of the Cauchy-Kowalevski procedure resulting in cross differences in spacetime but with some novel modifications which ensure the positivity of the discretization coefficients. It is then shown that the so-obtained spacetime cross differences result in changes to the wave speeds and can thus be incorporated within Roe or Steger-Warming schemes (with and without reconstruction-evolution) simply by altering the eigenvalues. The proposed approach shows advantages over alternatives in that it is positivity-preserving for the Euler equations. Further, it yields monotone solutions near discontinuities while exhibiting a truncation error in smooth regions less than the one of the second- or third-order accurate backward-difference-formula (BDF) for either small or large time steps. The high resolution and positivity preservation of the proposed discretization stencils are independent of the convergence acceleration technique which can be set to multigrid, preconditioning, Jacobian-free Newton-Krylov, block-implicit, etc. Thus, the current paper also offers the first implicit integration of the time-accurate Euler equations that is positivity-preserving in the strict sense (that is, the density and temperature are guaranteed to remain positive). This is in contrast to all previous positivity-preserving implicit methods which only guaranteed the positivity of the density, not of the temperature or pressure. Several stringent reacting and inert test cases confirm the positivity-preserving property of the proposed method as well as its higher resolution and higher computational efficiency over other second-order and third-order implicit temporal discretization strategies.

##### MSC:

76M12 | Finite volume methods applied to problems in fluid mechanics |

76N15 | Gas dynamics (general theory) |

76V05 | Reaction effects in flows |

##### Keywords:

implicit temporal discretization; monotonicity preservation; positivity preservation; dual time stepping; implicit schemes; reactive compressible flow
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##### References:

[1] | Payret, R.; Taylor, T., Computational methods for fluid flows, (1983), Springer NY |

[2] | Jameson, A., Time dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings, (1991), 10th Computational Fluid Dynamics Conference, Honolulu HI; AIAA Paper 91-1596 |

[3] | Venkateswaran, S.; Merkle, C. L., Dual time-stepping and preconditioning for unsteady computations, (33rd Aerospace Sciences Meeting and Exhibit, Reno, Nevada, USA, (1995)) |

[4] | DeRango, S.; Zingg, D. W., Improvements to a dual-time-stepping method for computing unsteady flows, AIAA J., 35, 9, 1548-1550, (1997) · Zbl 0900.76339 |

[5] | Pulliam, T. H.; Chaussee, D. S., A diagonal form of an implicit approximate-factorization algorithm, J. Comput. Phys., 39, 347-363, (1981) · Zbl 0472.76068 |

[6] | Beam, R.; Warming, R. F., An implicit finite-difference algorithm for hyperbolic systems in conservation-law form, J. Comput. Phys., 22, 1, 87-110, (1976) · Zbl 0336.76021 |

[7] | Beam, R.; Warming, R. F., An implicit factored scheme for the compressible Navier-Stokes equations, AIAA J., 16, 4, 393-402, (1978) · Zbl 0374.76025 |

[8] | Steger, J. L., Implicit finite-difference simulation of flow about arbitrary two-dimensional geometries, AIAA J., 16, 7, 679-686, (1978) · Zbl 0383.76013 |

[9] | Rumsey, C. L.; Sanetrik, M. D.; Biedron, R. T.; Melson, N. D.; Parlette, E. B., Efficiency and accuracy of time-accurate turbulent Navier-Stokes computations, Comput. Fluids, 25, 2, 217-236, (1996) · Zbl 0881.76067 |

[10] | Hansen, M. A.; Sutherland, J. C., Dual timestepping method for detailed combustion chemistry, Combust. Theory Model., 21, 2, 329-345, (2017) |

[11] | Sankaran, V.; Oefelein, J. C., Advanced preconditioning strategies for chemically reacting flows, (2007), paper AIAA-2007-1432, 45th AIAA Aerospace Sciences and Exhibit, Reno, NV, USA |

[12] | Hitch, B. D.; Lynch, E. D., Use of reduced, accurate ethylene combustion mechanisms for a hydrocarbon-fueled ramjet simulation, (2009), paper AIAA-2009-5384, 45th AIAA Joint Propulsion Conference and Exhibit, Denver, CO, USA |

[13] | Housman, J.; Barad, M.; Kiris, C.; Kwak, D., Space-time convergence analysis of a dual-time stepping method for simulating ignition overpressure waves, (Kuzmin, A., Computational Fluid Dynamics 2010, St-Petersburg, Russia, (2011), Springer), 645-652 · Zbl 1346.76132 |

[14] | Charest, M. R.J.; Groth, C. P.T., A high-order central ENO finite-volume scheme for three-dimensional turbulent reactive flows on unstructured mesh, (2013), paper AIAA-2013-2567, 21st AIAA Computational Fluid Dynamics Conference, San Diego, CA, USA |

[15] | Parent, B., Positivity-preserving flux-limited method for compressible fluid flow, Comput. Fluids, 44, 1, 238-247, (2011) · Zbl 1271.76216 |

[16] | Batten, P.; Leschziner, M. A.; Goldberg, U. C., Average-state Jacobians and implicit methods for compressible viscous and turbulent flows, J. Comput. Phys., 137, 38-78, (1997) · Zbl 0901.76043 |

[17] | Moryossef, Y.; Levy, Y., Unconditionally positive implicit procedure for two-equation turbulence models: application to \(k - \omega\) turbulence models, J. Comput. Phys., 220, 88-108, (2006) · Zbl 1158.76330 |

[18] | Mor-Yossef, Y.; Levy, Y., The unconditionally positive-convergent implicit time integration scheme for two-equation turbulence models: revisited, Comput. Fluids, 38, 1984-1994, (2009) · Zbl 1242.76179 |

[19] | Kuzmin, D., A guide to numerical methods for transport equations, (2010) |

[20] | Toro, E. F., Riemann solvers and numerical methods for fluid dynamics, (2009), Springer · Zbl 1227.76006 |

[21] | Huang, Y.; Lerat, A., Second-order upwinding through a characteristic time-step matrix for compressible flow calculations, J. Comput. Phys., 142, 445-472, (1998) · Zbl 0932.76052 |

[22] | Parent, B., Multidimensional flux difference splitting schemes, AIAA J., 53, 7, 1936-1948, (2015) |

[23] | Parent, B., Multidimensional high-resolution schemes for viscous hypersonic flows, AIAA J., 55, 1, 141-152, (2017) |

[24] | Parent, B., Positivity-preserving high-resolution schemes for systems of conservation laws, J. Comput. Phys., 231, 1, 173-189, (2012) · Zbl 1457.65060 |

[25] | Roe, P. L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43, 357-372, (1981) · Zbl 0474.65066 |

[26] | Parent, B., Positivity-preserving flux difference splitting schemes, J. Comput. Phys., 243, 1, 194-209, (2013) · Zbl 1349.76519 |

[27] | Dubroca, B., Positively conservative Roe’s matrix for Euler equations, (Proceedings of the 16th International Conference of Numerical Methods for Fluid Dynamics, Lecture Notes in Physics, vol. 515, (1998)), 272-277 |

[28] | Dubroca, B., Solveur de roe positivement conservatif, C. R. Acad. Sci., Ser. I Math., 329, 9, 827-832, (1999) · Zbl 0957.76049 |

[29] | Anderson, W. K.; Thomas, J. L.; Van Leer, B., Comparison of finite volume flux vector splittings for the Euler equations, AIAA J., 24, 1453-1460, (1986) |

[30] | Bardina, J.; Lombard, C. K., Three dimensional hypersonic flow simulations with the CSCM implicit upwind Navier-Stokes method, (1987), Proceedings of the 8th Computational Fluid Dynamics Conference, AIAA Paper 87-1114 |

[31] | MacCormack, R., A new implicit algorithm for fluid flow, (1997), paper AIAA-97-2100 |

[32] | Gerolymos, G.; Sénéchal, D.; Vallet, I., Analysis of dual-time-stepping for advection-diffusion-type equations with explicit subiterations, (2009), paper AIAA-2009-1608, 47th AIAA Aerospace Sciences Meeting, Orlando, FL, USA |

[33] | Dwight, R. P., Time-accurate Navier-Stokes calculations with approximately factored implicit schemes, (Groth, C.; Zingg, D. W., Computational Fluid Dynamics 2004, Toronto, ON, Canada, (2006), Springer), 211-217 |

[34] | Gottlieb, S.; Shu, C. W., Total variation diminishing Runge Kutta schemes, Math. Comput., 67, 73-85, (1998) · Zbl 0897.65058 |

[35] | Jiang, G.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126, 202-228, (1996) · Zbl 0877.65065 |

[36] | Briley, W. R.; McDonald, H., On the structure and use of linearized block implicit schemes, J. Comput. Phys., 34, 54-73, (1980) · Zbl 0436.76021 |

[37] | Parent, B.; Sislian, J., The use of domain decomposition in accelerating the convergence of quasi-hyperbolic systems, J. Comput. Phys., 179, 1, 140-169, (2002) · Zbl 1130.76399 |

[38] | Shu, C.-W.; Osher, S. J., Efficient implementation of essentially non-oscillatory shock capturing schemes II, J. Comput. Phys., 83, 32-78, (1989) · Zbl 0674.65061 |

[39] | Balsara, D. S.; Garain, S.; Shu, C. W., An efficient class of WENO schemes with adaptive order, J. Comput. Phys., 326, 780-804, (2016) · Zbl 1422.65146 |

[40] | McBride, B. J.; Zehe, M. J.; Gordon, S.; Glenn, N. A.S. A., Coefficients for calculating thermodynamic properties of individual species, (2002), NASA, TP 211556 |

[41] | Jachimowsky, C. J., An analytical study of the hydrogen-air reaction mechanism with application to scramjet combustion, (1988), NASA, TP 2791 |

[42] | Parent, B.; Macheret, S. O.; Shneider, M. N., Electron and ion transport equations in computational weakly-ionized plasmadynamics, J. Comput. Phys., 259, 51-69, (2014) · Zbl 1349.82151 |

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