Convergence and Accuracy of Numerical Methods for Trajectory Calculations

Abstract

Computation of trajectories by a kinematic method requires the numerical solution of the differential equation by which the trajectory is defined. A widely used method is the iterative scheme of Petterssen which has second-order accuracy. The convergence and accuracy of this scheme is investigated for some simple flow types where analytical solutions are available. The results are discussed in comparison to other methods, especially a method similar to the Petterssen scheme, which has been recommended for use in semi-Lagrangian advection schemes. The truncation error in trajectory calculations should be kept about one order of magnitude smaller than the total uncertainty, which is mainly due to analysis errors and limited resolution of the wind data. It is shown that for trajectory calculations based on the typical output of current numerical weather prediction models or comparable data, this requires a time step of about 15% of the time step necessary to achieve convergence. If a fixed time step is used, it should not exceed about 0.5 h. Flexible time steps, based on the estimate of the truncation error which is provided by the difference between the first and the second iteration, are an attractive alternative.