Accuracy of Finite-Difference Methods Applied to the Advection Equation

Numerical solutions to the equation for advection under conditions that permit an analytic solution are calculated using various finite-difference approximations, and their accuracy is investigated by comparison with the analytic solution. It is shown that upstream differencing introduces a pseudo-diffusive effect that is about as large as the effect of the turbulent diffusion modeled in typical small-scale circulation simulation; hence, the numerical solution is rendered inconsistent with the differential equation. In the solutions obtained with centered-difference schemes, an anomalous oscillation is present when the grid-spacing is too large to follow very closely the variations of the quantity being advected. This oscillation leads to inaccuracy and numerical instability. Of all the schemes investigated, only the Roberts-Weiss approximation advected the initial distribution correctly, but this scheme required about 10–40 times as much computer time as any of the other schemes.