The large time step (LTS) explicit scheme relaxes the Courant–Friedrichs–Lewy (CFL) restriction by propagating waves across multiple cells per time step and has shown clear efficiency gains for the homogeneous shallow water equations. Earlier Godunov-type LTS constructionscombined an exact Riemann solver with a multi-wave approximation of the rarefaction fan and a random choice method to suppress spurious oscillations. The present work proposes a simpler yet equally accurate variant. The star state is obtained from Toro’s closed-form two-rarefaction approximation (TRA) and then refined by a short Newton iteration; this combination is exact for two-rarefaction configurations, removes the systematic bias of TRA in two-shock configurations, and converges in two to three iterations from the TRA initial guess in all other cases. The rarefaction fan is split into sub-waves with linearly interpolated speeds, and each sub-wave is dispatched to either a left-moving or a right-moving propagator according to the sign of its mean speed. Fractional contributions are resolved based on a deterministic threshold with θ = 0.5. The scheme is validated based on seven one-dimensional flat-bottom Riemann problems covering dry-bed, wet-bed, two-rarefaction, and two-shock configurations, each at CFL numbers 0.9, 3, 8, and 15. The L1 error in water depth remains below 3% for all cases and is as low as 0.2% in the two-shock configuration at CFL = 15. The Xu 2014 dam break is reproduced with an error of 0.4% at CFL = 15 using only three time steps.
Huang et al. (Tue,) studied this question.