A Time-Splitting Scheme for the Elastic Equations Incorporating Second-Order Runge–Kutta Time Differencing

A forward-in-time splitting method for integrating the elastic equations is presented. A second-order Runge– Kutta time integrator (RK2) for the large-time-step integration is combined with the forward–backward scheme in a manner similar to the Klemp and Wilhelmson method. The new scheme produces fully second-order-accurate integrations for advection and gravity wave propagation. The RK2 scheme uses upwind discretizations for the advection terms and is easily combined with standard vertically semi-implicit techniques so as to improve computational efficiency when the grid aspect ratio becomes large. A stability analysis of the RK2 split-explicit scheme shows that it is stable for a wide range of advective and acoustic wave Courant numbers. The RK2 time-split scheme is used in a full-physics nonhydrostatic compressible cloud model. The implicit damping properties associated with the RK2’s third-order horizontal differencing allows for a significant reduction in the value of horizontal filtering applied to the momentum and pressure fields, while qualitatively the solutions appear to be better resolved than solutions from a leapfrog model.