Positivity-preserving flux difference splitting schemes.

*(English)*Zbl 1349.76519Summary: A positivity-preserving variant of the Roe flux difference splitting method is here proposed. Positivity-preservation is attained by modifying the Roe scheme such that the coefficients of the discretization equation become positive, with a coefficient considered positive if all its eigenvalues are positive and if its eigenvectors correspond to those of the flux Jacobian. Because the modification does not alter the wave speeds at the interface, the appealing attributes of the Roe flux difference splitting schemes are retained, such as high-resolution capture of discontinuous waves, low amount of artificial dissipation within viscous layers, and ease of convergence to steady-state. The proposed flux function is advantaged over previous positivity-preserving variants of the Roe method by being written in general matrix form and hence by being readily deployable to arbitrary systems of conservation laws. The stencils are extended to second-order accuracy through a newly-derived positivity-preserving total-variation-diminishing limiting process that is applied to the characteristic variables and that yields positive coefficients. Also derived is a positivity-preserving restriction on the time step for flux difference splitting schemes that is shown to depart significantly from the CFL condition in regions with high property gradients.

##### MSC:

76M20 | Finite difference methods applied to problems in fluid mechanics |

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

##### Keywords:

positivity preservation; rule of the positive coefficients; flux difference splitting (FDS); Roe solver; monotonicity preservation; total variation diminishing (TVD); centered TVD limiters; Yee-Roe scheme; interface averaging##### Software:

HLLE
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DOI

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##### References:

[1] | Roe, P. L., Approximate Riemann solvers, parameter vectors, and difference schemes, Journal of Computational Physics, 43, 357-372, (1981) · Zbl 0474.65066 |

[2] | Roe, P. L., Approximate Riemann solvers, parameter vectors, and difference schemes, Journal of Computational Physics, 135, 250-258, (1997) · Zbl 0890.65094 |

[3] | Godunov, S. K., A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics, Mathematics Sbornik, 47, 3, 271-306, (1959) · Zbl 0171.46204 |

[4] | Toro, E. F.; Spruce, M.; Speares, W., Restoration of the contact surface in the HLL-Riemann solver, Shock Waves, 4, 25-34, (1994) · Zbl 0811.76053 |

[5] | Edwards, J. R.; Liou, M.-S., Low-diffusion flux-splitting methods for flows at all speeds, AIAA Journal, 36, 1610-1617, (1998) |

[6] | Steger, J. L.; Warming, R. F., Flux vector splitting of the inviscid gasdynamics equations with application to finite difference methods, Journal of Computational Physics, 40, 263-293, (1981) · Zbl 0468.76066 |

[7] | Harten, A.; Lax, P. D.; Van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Review, 25, 1, 35-61, (1983) · Zbl 0565.65051 |

[8] | Einfeldt, B.; Munz, C. D.; Roe, P. L.; Sjogreen, B., On Godunov-type methods near low densities, Journal of Computational Physics, 92, 273-295, (1991) · Zbl 0709.76102 |

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

[10] | Dubroca, B., Solveur de roe positivement conservatif, Comptes Rendus de l’Académie des Sciences - Series I - Mathematics, 329, 9, 827-832, (1999) · Zbl 0957.76049 |

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

[12] | Parent, B., Positivity-preserving high-resolution schemes for systems of conservation laws, Journal of Computational Physics, 231, 1, 173-189, (2012) · Zbl 1457.65060 |

[13] | Yee, H. C.; Klopfer, G. H.; Montagné, J.-L., High-resolution shock-capturing schemes for inviscid and viscous hypersonic flows, Journal of Computational Physics, 88, 31-61, (1990) · Zbl 0697.76079 |

[14] | Yee, H. C., Construction of explicit and implicit symmetric TVD schemes and their applications, Journal of Computational Physics, 68, 151-179, (1987) · Zbl 0621.76026 |

[15] | H.C. Yee, A Class of High-Resolution Explicit and Implicit Shock Capturing Methods, TM 101088, NASA, 1989. |

[16] | Parent, B.; Sislian, J. P., Validation of wilcox \(k \omega\) model for flows characteristic to hypersonic airbreathing propulsion, AIAA Journal, 42, 2, 261-270, (2004) |

[17] | Noh, W. F., Errors for calculations of strong shocks using an artificial viscosity and an artificial heat flux, Journal of Computational Physics, 72, 78-120, (1987) · Zbl 0619.76091 |

[18] | Johnsen, E.; Larsson, J.; Bhagatwala, A. V.; Cabot, W. H.; Moin, P.; Olson, B. J.; Rawat, P. S.; Shankar, S. K.; Sjogreen, B.; Yee, H. C.; Zhong, X.; Lele, S. K., Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves, Journal of Computational Physics, 229, 1213-1237, (2010) · Zbl 1329.76138 |

[19] | J.R. Kamm, F.X. Timmes, On Efficient Generation of Numerically Robust Sedov Solutions, LA-UR 07-2849, Los Alamos National Laboratory, 2007. |

[20] | Maire, P.-H., A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes, Journal of Computational Physics, 228, 2391-2425, (2009) · Zbl 1156.76434 |

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