Over-relaxation applied to the MacCormack finite-difference scheme

An over-relaxation procedure is applied to the MacCormack finite-difference scheme in order to reduce the computation time required to obtain a steady-state solution. The implementation of this acceleration procedure to an existing computer program using the regular MacCormack method is extremely simple and does not require additional storage. The over-relaxation procedure does not alter the steady-state solution, which is second-order accurate. The method is first applied to Burgers' equation. A stability condition and an expression for the increase in the rate of convergence are derived. The method is then applied to the calculation of the hypersonic viscous flow over a flat plate, using the complete Navier-Stokes equations, and the inviscid flow over a wedge. Reductions in computing time by factors of 3 and 1.5, respectively, are obtained by over-relaxation.