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**Entropy splitting and numerical dissipation.**
*(English)*
Zbl 0987.65086

This paper addresses some computational aspects of an existing arbitrary order central difference scheme based on flux entropy splitting for systems of hyperbolic conservation laws. In particular, the authors first investigate the choice of the arbitrary parameter which determines the amount of splitting for the problem of a perfect gas. The choice of the parameter is problem dependent. The authors then investigate the influence of the splitting on the nonlinear stability of the central difference scheme for long time integrations of unsteady flows. The paper also investigate the influence of the splitting on the numerical dissipation if such a dissipation is needed to stabilize the central scheme. The techniques for the equations governing a perfect gas are extended to problems with other equation states such as a thermally perfect gas and magnetohydrodynamics. Extensive numerical experiments are performed to demonstrate the effectiveness of the techniques developed.

Reviewer: Song Wang (Nedlands)

### MSC:

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

76N15 | Gas dynamics (general theory) |

76W05 | Magnetohydrodynamics and electrohydrodynamics |

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

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |

35L65 | Hyperbolic conservation laws |

### Keywords:

entropy variables; systems of hyperbolic conservation laws; thermally perfect gas; flux entropy splitting; central difference scheme; compact schemes; nonlinear stability; shock-capturing methods; nonlinear filters; conservative differencing; unsteady flows; magnetohydrodynamics; numerical experiments
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\textit{H. C. Yee} et al., J. Comput. Phys. 162, No. 1, 33--81 (2000; Zbl 0987.65086)

### References:

[1] | Kreiss, H.-O.; Scherer, G., On the existence of energy estimates for difference approximations for hyperbolic systems, (1977) |

[2] | Strand, B., Summation by parts for finite difference approximations for d/dx, J. comput. phys., 110, 47, (1994) · Zbl 0792.65011 |

[3] | Olsson, P., Summation by parts, projections and stability, Math. comp., 64, 1035, (1995) · Zbl 0828.65111 |

[4] | Gustafsson, B.; Olsson, P., Fourth-order difference method for hyperbolic ibvps, J. comput. phys., 1177, 300, (1995) · Zbl 0823.65083 |

[5] | Olsson, P.; Oliger, J., Energy and maximum norm estimates for nonlinear conservation laws, (1994) |

[6] | Olsson, P., Summation by parts, projections and stability, II, Math. comp., 64, 212, (1995) |

[7] | Olsson, P., Summation by parts, projections and stability, III, (1995) · Zbl 0848.65064 |

[8] | Gerritsen, M., Designing an efficient solution strategy for fluid flows, (October 1996) |

[9] | Gerritsen, M.; Olsson, P., Designing an efficient solution strategy for fluid flows. I. A stable high order finite difference scheme and sharp shock resolution for the Euler equations, J. comput. phys., 129, 245, (1996) · Zbl 0899.76281 |

[10] | Harten, A., On the symmetric form of systems for conservation laws with entropy, (1981) |

[11] | Harten, A., J. comput. phys., 49, 151, (1983) |

[12] | Steger, J.L.; Warming, R.F., Flux vector splitting of inviscid gas dynamics equations with applications to finite difference methods, J. comput. phys., 40, 263, (1981) · Zbl 0468.76066 |

[13] | Hughes, R.J.R.; Franca, L.P.; Mallet, M., A new finite element formulation for computational fluid dynamics. I. symmetric forms of the compressible Euler and navier – stokes equations and the second law of thermodynamics, Comput. methods appl. mech. eng., 54, 223, (1986) · Zbl 0572.76068 |

[14] | Yee, H.C.; Sandham, N.D.; Djomehri, M.J., Low dissipative high order shock-capturing methods using characteristic-based filters, (May 1998) |

[15] | Yee, H.C.; Sandham, N.D.; Djomehri, M.J., J. comput. phys., 150, 199, (1999) |

[16] | Harten, A., The artificial compression method for computation of shocks and contact discontinuities. III. self-adjusting hybrid schemes, Math. comp., 32, 363, (1978) · Zbl 0409.76057 |

[17] | Sjogreen, B.; Yee, H.C., Multi-resolution wavelet based adaptive numerical dissipation control for shock-turbulence computations, (2000) |

[18] | Sandham, N.D.; Yee, H.C., Entropy splitting for high order numerical simulation of compressible turbulence, (2000) · Zbl 1139.76332 |

[19] | N. D. Sandham, and, H. C. Yee, Proceedings of the 1st International Conference on CFD, July 10-14, 2000, Kyoto, Japan. |

[20] | Hadjadj, A.; Yee, H.C.; Sandham, N.D., Comparison of WENO with low dissipative high order schemes for compressible turbulence computations, (2000) |

[21] | Vinokur, M.; Yee, H.C., Extension of efficient low dissipative high order schemes for 3-D curvilinear moving girds, (2000) |

[22] | M. Vinokur, and, H. C. Yee, Proceedings of the Computing the Future III: Frontier of CFD, June 26-28, 2000, Half Moon Bay, CA. |

[23] | Yee, H.C.; Vinokur, M.; Djomehri, M.J., Entropy splitting and numerical dissipation, (1999) · Zbl 0987.65086 |

[24] | Dahlquist, G., Some properties of linear multistep and one-leg methods for ordinary differential equations, Conference proceeding, 1979 SIGNUM meeting on numerical ODE’s, champaign, IL, (1979) · Zbl 0436.65048 |

[25] | Yee, H.C.; Harten, A., Implicit TVD schemes for hyperbolic conservation laws in curvilinear coordinates, (July 1985) |

[26] | Yee, H.C.; Harten, A., Aiaa j., 25, 266, (1987) |

[27] | Yee, H.C., On symmetric and upwind TVD schemes, Proceedings of the 6th GAMM-conference on numerical methods in fluid mechanics, gottingen, Germany, (September 1985) |

[28] | Yee, H.C., Linearized form of implicit TVD schemes for multidimensional Euler and navier – stokes equations, Comput. math. appl. A, 12, 413, (1986) · Zbl 0597.76028 |

[29] | Yee, H.C., Construction of explicit and implicit symmetric TVD schemes and their applications, J. comput. phys., 68, 151, (1987) · Zbl 0621.76026 |

[30] | H. C. Yee, A Class of High-Resolution Explicit and Implicit Shock-Capturing Methods, VKI Lecture Series 1989-04, March 6-10, 1989; also NASA TM-101088, February 1989. |

[31] | Yee, H.C.; Klopfer, G.H.; Montagne, J.-L., High-resolution shock-capturing schemes for inviscid and viscous hypersonic flows, J. comput. phys., 88, 31, (1990) · Zbl 0697.76079 |

[32] | Harten, A., A high resolution scheme for computation of weak solutions of hyperbolic conservation laws, J. comput. phys., 49, 35, (1983) |

[33] | Beam, R.M.; Warming, R.F., An implicit finite-difference algorithm for hyperbolic systems in conservation law form, J. comput. phys., 22, 87, (1976) · Zbl 0336.76021 |

[34] | Vichnevetsky, R., Numerical filtering for partial differencing equations, (1974) |

[35] | Alpert, P., Implicit filtering in conjunction with explicit filtering, J. comput. phys., 44, 212, (1981) · Zbl 0492.76054 |

[36] | Lele, S.A., Compact finite difference schemes with spectral-like resolution, J. comput. phys., 103, 16, (1992) · Zbl 0759.65006 |

[37] | Visbal, M.R.; Gaitonde, D.V., High-order accurate methods for unsteady vortical flows on curvilinear meshes, (1998) |

[38] | Gaitone, D.V.; Visbal, M.R., Further development of a navier – stokes solution procedure based on higher-order formulas, (1999) |

[39] | Vinokur, M., An analysis of finite-difference and finite-volume formulations of conservation laws, J. comput. phys., 81, 1, (1989) · Zbl 0662.76039 |

[40] | Abgrall, R., Extension of Roe’s approximate Riemann solver to equilibrium and nonequilibrium flows, Notes numer. fluid mech., 29, 1, (1990) |

[41] | Spekreijse, R.; Hagmeijer, R., Derivation of a roe scheme for an N-species chemically reacting gas in thermal equilibrium, Notes numer. fluid mech., 29, 522, (1989) |

[42] | Chalot, F.; Hughes, T.J.R.; Shakib, F., Symmetrization of conservation laws with entropy for high-temperature hypersonic computations, Comput. systems eng., 1, 465, (1990) |

[43] | Roe, P.L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. comput. phys., 43, 357, (1981) · Zbl 0474.65066 |

[44] | Carpenter, M.H.; Gottlieb, D.; Abarbanel, S.; Don, W.-S., The theoretical accuracy of runge – kutta time discretizations for initial value problem: A study of the boundary error, SIAM J. sci. comput., 16, 1241, (1995) · Zbl 0839.65098 |

[45] | Harten, A.; Hyman, J.M., A self-adjusting grid for the computation of weak solutions of hyperbolic conservation laws, J. comput. phys., 50, 235, (1983) · Zbl 0565.65049 |

[46] | Sandham, N.D.; Reynolds, W.C., A numerical investigation of the the compressible mixing layer, (1989) · Zbl 0717.76094 |

[47] | T. Lumpp, Compressible mixing layer computations with high-order ENO schemes, in, 15th Intl. Conf. on Num. Meth. in Fluid Dynamics, Monterey, June 1996. |

[48] | Fu, D.; Ma, Y., A high order accurate difference scheme for complex flow fields, J. comput. phys., 134, 1, (1997) · Zbl 0882.76054 |

[49] | Sandham, N.D.; Yee, H.C., Performance of low dissipative high order TVD schemes for shock-turbulence interactions, (April 1998) |

[50] | Sandham, N.D.; Yee, H.C., A numerical study of a class of TVD schemes for compressible mixing layers, (May 1989) |

[51] | Yee, H.C.; Sandham, N.D.; Sjogreen, B.; Hadjadj, A., Progress in the development of a class of efficient low dissipative high order shock-capturing methods, (2000) |

[52] | H. C. Yee, N. D. Sandham, B. Sjogreen, and, A. Hadjadj, Proceedings of the Symposium on CFD for the 21st Century, July 15-17, 2000, Kyoto, Japan. |

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