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**Construction of explicit and implicit symmetric TVD schemes and their applications.**
*(English)*
Zbl 0621.76026

A one-parameter family of second-order explicit and implicit total variation diminishing (TVD) schemes is reformulated so that a simplier and wider group of limiters is included. The resulting scheme can be viewed as a symmetrical algorithm with a variety of numerical dissipation terms that are designed for weak solutions of hyperbolic problems. This is a generalization of recent works of P. L. Roe [ICASE Report No.84-53 (Oct. 1984)] and S. F. Davis [ICASE Report No.84-20 (June 1984)] to a wider class of symmetric schemes other than Lax-Wendroff. The main properties of the present class of schemes are that they can be implicit, and, when steady-state calculations are sought, the numerical solution is independent of the time step. Numerical experiments with two- dimensional unsteady and steady-state airfoil calculations show that the proposed symmetric TVD schemes are quite robust and accurate.

### MSC:

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

76M99 | Basic methods in fluid mechanics |

65Z05 | Applications to the sciences |

### Keywords:

total variation diminishing schemes; one-parameter family; limiters; symmetrical algorithm; numerical dissipation; weak solutions; hyperbolic problems; steady-state calculations; steady-state airfoil### References:

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