On Forward-in-Time Differencing for Fluids: Extension to a Curvilinear Framework

This paper extends the discussion of fully second-order-accurate, forward-in-time, finite-difference schemes for the advection equation with arbitrary forcing (which is viewed as a prototype for the prognostic equations of fluid dynamics) to an arbitrary curvilinear system of coordinates. Since forward-in-time schemes derive ultimately from Taylor series analysis of the uncentered-in-time differencing, it is important to include the appropriate metric terms explicitly into the algorithm's design. A rigorous truncation-error analysis leads to a compact scheme that preserves (to second-order accuracy) the consistency of Eulerian and Lagrangian formulations for fluids. Alternative approximations to the advective velocity in the transport flux are also discussed. In order to achieve second-order accuracy of the forward-in-time approximation, the advective velocity must be evaluated to at least first-order accuracy at the intermediate time level. Such a temporal staggering is usually simulated by means of either linear interpolation or linear extrapolation. The alternative considered in this paper employs an extrapolation consistent with the governing equations of motion. This approximation is derived from concepts inherent in Runge-Kutta methods for ordinary differential equations. It allows at least twice the usual time step in simulations of elastic systems, where high-speed propagating modes dominate the computational stability. These theoretical considerations are illustrated with idealized tests and examples of shallow-water flows on the sphere.