Linearized form of implicit TVD schemes for the multidimensional Euler and Navier-Stokes equations.

*(English)*Zbl 0597.76028Summary: Linearized alternating direction implicit (ADI) forms of a class of total variation diminishing (TVD) schemes for the Euler and Navier-Stokes equations have been developed. These schemes are based on the second- order-accurate TVD schemes for hyperbolic conservation laws developed by A. Harten [e.g.: SIAM J. Numer. Anal. 21, 1-23 (1984; Zbl 0547.65062)].

They have the property of not generating spurious oscillations across shocks and contact discontinuities. In general, shocks can be captured within 1-2 grid points. These schemes are relatively simple to understand and easy to implement into a new or existing computer code. One can modify a standard three-point central-difference code by simply changing the conventional numerical dissipation term into the one designed for the TVD scheme. For steady-state applications, the only difference in computation is that the current schemes require a more elaborate dissipation term for the explicit operator; no extra computation is required for the implicit operator. Numerical experiments with the proposed algorithms on a variety of steady-state airfoil problems illustrate the versatility of the schemes.

They have the property of not generating spurious oscillations across shocks and contact discontinuities. In general, shocks can be captured within 1-2 grid points. These schemes are relatively simple to understand and easy to implement into a new or existing computer code. One can modify a standard three-point central-difference code by simply changing the conventional numerical dissipation term into the one designed for the TVD scheme. For steady-state applications, the only difference in computation is that the current schemes require a more elaborate dissipation term for the explicit operator; no extra computation is required for the implicit operator. Numerical experiments with the proposed algorithms on a variety of steady-state airfoil problems illustrate the versatility of the schemes.

##### MSC:

76D05 | Navier-Stokes equations for incompressible viscous fluids |

76M99 | Basic methods in fluid mechanics |

##### Keywords:

Linearized alternating direction implicit (ADI) forms; total variation diminishing (TVD) schemes; Navier-Stokes equations; hyperbolic conservation laws; spurious oscillations; contact discontinuities; three- point central-difference code; numerical dissipation; steady-state applications; steady-state airfoil problems
PDF
BibTeX
XML
Cite

\textit{H. C. Yee}, Comput. Math. Appl., Part A 12, 413--432 (1986; Zbl 0597.76028)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Harten, A.; Harten, A., A high resolution scheme for the computation of weak solutions of hyperbolic conservation laws, NYU report, J. comp. phys., 49, 357-393, (1983) · Zbl 0565.65050 |

[2] | Harten, A.; Harten, A., On a class of high resolution total-variation-stable finite-difference schemes, NYU report, SIAM J. numer. anal., 21, 1-23, (1984) · Zbl 0547.65062 |

[3] | Yee, H.C., On the implementation of a class of upwind schemes for system of hyperbolic conservation law, NASA technical memorandum 86839, (Sept. 1985) |

[4] | Yee, H.C.; Warming, R.F.; Harten, A.; Yee, H.C.; Warming, R.F.; Harten, A., Implicit total variation diminishing (TVD) schemes for steady-state calculations, (), J. comp. phys., 57, 327-360, (1985) · Zbl 0631.76087 |

[5] | Pulliam, T.H.; Barton, J.T., Euler computations of AGARD working group 07 airfoil test cases, AIAA paper no. 85-0018, (1985) |

[6] | Huang, L.C., Psuedo-unsteady difference schemes for discontintuous solutions of steady-state, one-dimensional fluid dynamics problems, J. comp. phys., 42, 195-211, (1981) |

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

[8] | Harten, A.; Hyman, J.M., A self-adjusting grid for the computation of weak solutions of hyperbolic conservation laws, J. comp. phys., 50, 235-269, (1983) · Zbl 0565.65049 |

[9] | Sweby, P.K., High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. num. analy., 21, 995-1011, (1984) · Zbl 0565.65048 |

[10] | Roe, P.L., Some contributions to the modelling of discontinuous flows, () · Zbl 0741.65077 |

[11] | Harten, A.; Osher, S., Uniformly high order nonoscillatory schemes, UCLA math. report, (1985) |

[12] | Yee, H.C.; Warning, R.F.; Harten, A., Application of TVD schemes for the Euler equations of gas dynamics, () · Zbl 0526.76080 |

[13] | Goodman, J.B.; LeVeque, R.J., On the accuracy of stable schemes for 2D scalar conservation laws, NYU report, (May 1983), New York |

[14] | Yee, H.C.; Kutler, P., Application of second-order-accurate total variation diminishing (TVD) schemes to the Euler equations in general geometries, Nasa tm-85845, (August 1983) |

[15] | Yee, H.C.; Harten, A., Implicit TVD schemes for hyperbolic conservation laws in curvilinear coordinates, () · Zbl 0631.76087 |

[16] | Yee, H.C., On symmetric and upwind TVD schemes, Proceedings of the 6th GAMM conference on numerical methods in fluid mechanics, NASA technical memorandum 86842, (Sept. 1985) |

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

[18] | Pulliam, T.H.; Steger, J., Recent improvements in efficiency, accuracy and convergence for implicit approximate factorization algorithms, AIAA paper no. 85-0360, (1985) |

[19] | Jameson, A., Numerical solutions of the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes, AIAA paper no. 81-1259, (1981) |

[20] | Baldwin, B.; Lomax, H., Thin layer approximation and algebraic model for separated turbulent flows, AIAA paper no. 78-257, (1978) |

[21] | MacCormack, R.W., A rapid solver for hypervolic systems of equations, () · Zbl 0493.76068 |

[22] | Cook, P.H., Aerofoil RAE2822â€”pressure distributions and boundary layer and wake measurements, Agard-ar-138, (1979) |

[23] | Mehta, A comparison of interactive boundary layer and thin-layer Navier-Stokes procedures, () · Zbl 0598.76056 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.