Numerical Methods for the Compressible Navier-Stokes Equations Using an Explicit Time-Marching Technique: (Comparison between the Four Stage Runge-Kutta and Two Stage Rational Runge-Kutta Schemes)

The four stage Runge-Kutta(4SRK) and two stage Rational Runge-Kutta(2SRRK) schemes have been applied to solve the compressible Navier-Stokes equations, in order to develop a fast, accurate explicit time-marching technique suitable for vectorization. On a supercomputer, a problem of shock-boundary layer interaction is calculated by use of these schemes combined with local timestepping. It is shown that the 4SRK scheme with the fourth-order central differencing of convective terms is stable out to a Courant number of 2.06 according to Neumann's stability criterion. The practical limit of Courant number has been very close to the theoretical limit. At a high Reynolds number, the Courant number limit of 2SRRK scheme is significantly less than that of conventional explicit methods. Both the 4SRK and 2SRRK schemes are readily vectorizable. The use of implicit residual averaging reduces iterations, but is not suitable for vectorization.