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Linearized form of implicit TVD schemes for the multidimensional Euler and Navier-Stokes equations. (English) Zbl 0597.76028
Summary: 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.

MSC:
76D05 Navier-Stokes equations for incompressible viscous fluids
76M99 Basic methods in fluid mechanics
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