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

We examine the application of a new implicit unconditionallystable high-resolution TVD scheme to steady-state calculations. It is a member of a one-parameter family of explicit and implicit second-order accurate schemes developed by Harten 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 highlyresolved approximation to the steady-state solution. A detailed implementation of the implicit scheme for the oneand two-dimensional compressible inviscid equations of gas dynamics is presented. Some numerical computations of oneand two-dimensional fluid flows containing shocks demonstrate the efficiency and accuracy of this new scheme. §