Construction of explicit and implicit symmetric TVD schemes and their applications (Q1090539)

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 \textit{P. L. Roe} [ICASE Report No.84-53 (Oct. 1984)] and \textit{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.