Implicit total variation diminishing (TVD) schemes for steady-state calculations.

*(English)*Zbl 0631.76087The application of a new implicit unconditionally stable high-resolution TVD scheme to steady-state calculations is examined. It is a member of a one-parameter family of explicit and implicit second-order accurate schemes developed by the third author [ibid. 49, 357-393 (1983; Zbl 0565.65050)] for the computation of weak solutions of one-dimensional hyperbolic conservation laws. This scheme is guaranteed not to generate spurious oscillations for a nonlinear scalar equation and a constant coefficient system. Numerical experiments show that this scheme not only has a fairly rapid convergence rate, but also generates a highly resolved approximation to the steady-state solution. A detailed implementation of the implicit scheme for the one- and two-dimensional compressible inviscid equations of gas dynamics is presented. Some numerical computations of one- and two-dimensional fluid flows containing shocks demonstrate the efficiency and accuracy of this new scheme.

##### MSC:

76N15 | Gas dynamics, general |

76L05 | Shock waves and blast waves in fluid mechanics |

76M99 | Basic methods in fluid mechanics |

##### Keywords:

implicit unconditionally stable high-resolution TVD scheme; steady-state calculations; explicit and implicit second-order accurate schemes; weak solutions of one-dimensional hyperbolic conservation laws; spurious oscillations; rapid convergence; highly resolved approximation to the steady-state solution; two-dimensional compressible inviscid equations of gas dynamics; two-dimensional fluid flows
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##### References:

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