Analysis of Semi-Lagrangian advection using finite difference methods

Abstract

The advective terms in several numerical weather models have been integrated with Semi-Lagrangian methods for decades. Their stability allows for larger time steps than explicit finite difference methods, while experiencing less dispersion than implicit finite difference methods. In the case of the one dimensional advection equation, it turns out that for uniform and constant velocity, the Semi-Lagrangian method is equivalent to a shifted finite difference method. The degree of polynomial interpolation determines the truncation of the Taylor series and accuracy of the finite difference approximations. The shift is necessary to account for arbitrarily large time steps without losing stability as occurs for explicit finite difference methods. For linear interpolation, the integration is guarenteed to be stable for non-uniform and non-constant velocity, hile higher order interpolation can in theory experience instability. The instabilities occasionally observed in the HARMONIE operational code are a part of the departure point problem. This thesis proves the existence and uniqueness of and convergence to these departure points. In areas with large vertical velocity gradients, the convergence of fixed point iteration can be too slow, resulting in nonphysical crossing of characteristics and therefore negative model layer heights. This could be the key to finding a criterion to prevent runtime errors occurring and improving reliability of numerical weather prediction.