Effects of Using a Posteriori Methods for the Conservation of Integral Invariants

Abstract

An examination is made on the nature and effect of using a posteriors adjustments on nonconservative finite-difference schemes to enforce integral invariants of the corresponding analytic system. Using the one-dimensional linear advection equation and the first-order forward in time and upstream in space scheme, it is shown that using an a posteriors constraint restoration technique conserves the globally integrated mass and energy of the system by falsely generating energy in scales which survive the numerical integration. Next, the restoration technique is applied to the nonlinear shallow water system on a sphere using a fourth-order accurate model on a nonstaggered A-grid, which analytically conserves mass, total energy, and momentum but needs periodic global filtering to control nonlinear instability due to aliasing. Using a modification of the a posteriors constraint restoration technique to ensure global conservation of mass, energy and potential enstrophy, nearly perfect conservation is obtained during a 15-day forecast. Forecast skill, however, does not improve in the sense that the A-grid model fails to simulate well the flow over a single mountain (when run at a coarse resolution of 8° lat.?10° long.) when compared with forecasts made using an implicitly conserving model for mass, energy and potential enstrophy on a staggered C-grid at the same resolution. In addition, it is shown that a posteriori conservation of potential enstrophy does not restrict the cascade of energy into short scales as does implicit potential enstrophy conservation. Both models performed well at 4° ? 5° resolution, conserving, all global invariants to a high degree even without the a posteriori restoration process.