Abstract
With a finite volume approach, a flux-form semi-Lagrangian (TFSL) scheme with space-time transformation was developed to provide stable and accurate algorithm in solving the advection-diffusion equation. Different from the existing fluxform semi-Lagrangian schemes, the temporal integration of the flux from the present to the next time step is transformed into a spatial integration of the flux at the side of a grid cell (space) for the present time step using the characteristic-line concept. The TFSL scheme not only keeps the good features of the semi-Lagrangian schemes (no Courant number limitation), but also has higher accuracy (of a second order in both time and space). The capability of the TFSL scheme is demonstrated by the simulation of the equatorial Rossby-soliton propagation. Computational stability and high accuracy makes this scheme useful in ocean modeling, computational fluid dynamics, and numerical weather prediction.
Highlights
From a physical point of view, advection of a passive tracer is the simple transition of a quantity without diffusion and dispersion
As applied to a constituent advection problem, these numerical artifacts manifest themselves as nonphysical mixing by numerical diffusion, nonphysical highs and lows in the constituent field caused by dispersion, and nonphysical tracer spectra caused by trapping in nonpropagating small spatial scales (Rood 1987)
The stability and accuracy of numerical schemes for ocean models are usually verified using the propagation of a Rossby soliton on an equatorial beta-plane
Summary
From a physical point of view, advection of a passive tracer is the simple transition of a quantity without diffusion and dispersion. Numerical approaches in atmospheric and oceanic modeling inevitably introduce diffusion (or dissipation) and dispersion into the approximate solution. The stability and accuracy of numerical schemes for ocean models are usually verified using the propagation of a Rossby soliton on an equatorial beta-plane. To show the benefit of using the TFSL scheme, we first show instability and large diffusion and dispersion errors in numerical solution of the Rossby soliton using the existing schemes such as the flux-form upwind, flux-form central, Lax-Wendroff, and flux-form semi-Lagrangian schemes.
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