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

Abstract

Linearized alternating direction implicit (ADI) forms of a class of total vvariation 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 Harten[1,2]. 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-differences 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.