A modified leapfrog scheme for shallow water equations

In the 1D linearized shallow water equations, the Courant number should be <0.5 for stability in the original Leapfrog (LF) scheme. Here, we propose using the time-averaged height in the pressure gradient force in the momentum equations. The stability analysis shows that the new scheme is neutral when Courant number <1. The scheme is 2nd order accurate in both time and space. It does not require iterations and can be easily applied in 2D or 3D wave equations. The numerical simulations for 2-D linearized shallow water equations are consistent with those obtained from a 2-time-step semi-implicit scheme. The time filter can also be easily applied to this modified LF scheme.

► We propose modified leapfrog scheme using time-averaged height in pressure gradient. ► New scheme has the same accuracy but twice of efficiency as the original LF scheme. ► The numerical results are verified by a new modified forward–backward scheme.