Abstract. The accuracy of trajectory calculations performed by Lagrangian particle dispersion models (LPDMs) depends on various factors. The optimization of numerical integration schemes used to solve the trajectory equation helps to maximize the computational efficiency of large-scale LPDM simulations. We analyzed global truncation errors of six explicit integration schemes of the Runge–Kutta family, which we implemented in the Massive-Parallel Trajectory Calculations (MPTRAC) advection module. The simulations were driven by wind fields from operational analysis and forecasts of the European Centre for Medium-Range Weather Forecasts (ECMWF) at T1279L137 spatial resolution and 3 h temporal sampling. We defined separate test cases for 15 distinct regions of the atmosphere, covering the polar regions, the midlatitudes, and the tropics in the free troposphere, in the upper troposphere and lower stratosphere (UT/LS) region, and in the middle stratosphere. In total, more than 5000 different transport simulations were performed, covering the months of January, April, July, and October for the years 2014 and 2015. We quantified the accuracy of the trajectories by calculating transport deviations with respect to reference simulations using a fourth-order Runge–Kutta integration scheme with a sufficiently fine time step. Transport deviations were assessed with respect to error limits based on turbulent diffusion. Independent of the numerical scheme, the global truncation errors vary significantly between the different regions. Horizontal transport deviations in the stratosphere are typically an order of magnitude smaller compared with the free troposphere. We found that the truncation errors of the six numerical schemes fall into three distinct groups, which mostly depend on the numerical order of the scheme. Schemes of the same order differ little in accuracy, but some methods need less computational time, which gives them an advantage in efficiency. The selection of the integration scheme and the appropriate time step should possibly take into account the typical altitude ranges as well as the total length of the simulations to achieve the most efficient simulations. However, trying to summarize, we recommend the third-order Runge–Kutta method with a time step of 170 s or the midpoint scheme with a time step of 100 s for efficient simulations of up to 10 days of simulation time for the specific ECMWF high-resolution data set considered in this study. Purely stratospheric simulations can use significantly larger time steps of 800 and 1100 s for the midpoint scheme and the third-order Runge–Kutta method, respectively.
Stability of the Shallow Axisymmetric Parabolic-Conic Bimetallic Shell by Nonlinear Theory