Iterative implementation of an implicit - explicit hybrid scheme for solving the Euler equations is described in this paper. The scheme was proposed by Fryxell et al. (J. Comput. Phys. 63, 283 (1986)), is of the Godunov-type in both explicit and implicit regimes, is conservative, and is accurate to second order in both space and time for all Courant numbers. Only a single level of iterations is involved in the implementation, which solves both the implicit relations arising from upstream centered differences for all wave families and the nonlinearity of the Euler equations. The number of iterations required to reach a converged solution may be significantly reduced by the introduction of the multicolors proposed in this paper. Only a small number of iterations are needed in the scheme for a simulation with large time steps. The multicolors may also be applied to other linear and nonlinear wave equations for numerical solutions.
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The authors wish to thank B. Fyxell, and Wenlong Dai wishes to thank H. Chen, for the discussion about implicit schemes. This work was supported by the U.S. National Science Foundation under Grant NSF-ASC-9309829, by the Minnesota Supercomputer Institute, and by the Army Research Office contract number DAALO3-89-C-0038 with Army High Performance Computing Research Center at the University of Minnesota.