This paper studies the sweep direction effect on the convergence rate and CPU time of the implicit unfactored Gauss-Seidel line relaxation (GSLR) method for compressible flows. The line Gauss-Seidel iteration is also compared with the LU-SGS (lower-upper symmetric Gauss-Seidel) method. A modified LU-SGS method, namely LU-GSLR, is studied. The LU-GSLR use the unfactored GSLR method with the simple matrix of the LU-SGS, The numerical experiments indicate that for the external flows, the line Gauss-Seidel relaxation methods with sweeping in all directions achieves the optimum convergence rate and CPU efficiency. For an inviscid transonic internal flow, the best convergence rate is obtained with sweeping in streamwise direction only. Within each time step, one sweep (a forward sweep plus a backward sweep) per time step is sufficient. For the three implicit methods, GSLR, LU-SGS, and LU-GSLR, the GSLR is the most efficient method when the Roe scheme is used. The LU-GSLR method is a feasible method and can achieve better efficiency than the LU-SGS for some cases.
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