The Application of the Turkel–Zwas Explicit Large Time-Step Scheme to a Hemispheric Barotropic Model with Constraint Restoration

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Abstract
The Turkel–Zwas (T–Z) explicit large time-step scheme addresses the issue of fast and slow time scales in shallow-water equations by treating terms associated with fast waves on a coarser grid but to a higher accuracy than those associated with the slow-propagating Rossby waves. The T–Z scheme has been applied for solving the shallow-water equations on a fine-mesh hemispheric domain, using realistic initial conditions and an increased time step. To prevent nonlinear instability due to nonconservation of integral invariants of the shallow-water equations in long-term integrations, we enforced a posteriori their conservation. Two methods, designed to enforce a posteriori the conservation of three discretized integral invariants of the shallow-water equations, i.e., the total mass, total energy and potential enstrophy, were tested. The first method was based on an augmented Lagrangian method (Navon and de Villiers), while the second was a constraint restoration method (CRM) due to Miele et al., satisfying the requirement that the constraints be restored with the least-squares change in the field variables. The second method proved to be simpler, more efficient and far more economical with regard to CPU time, as well as easier to implement for first-time users. The CRM method has been proven to be equivalent to the Bayliss–Isaacson conservative method. The T–Z scheme with constraint restoration was run on a hemispheric domain for twenty days with no sign of impending numerical instability and with excellent conservation of the three integral invariants. Time steps approximately three times larger than allowed by the explicit CFL condition were used. The impact of the larger time step on accuracy is also discussed.

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