ISSN:
1436-5065
Source:
Springer Online Journal Archives 1860-2000
Topics:
Geography
,
Physics
Notes:
Summary Two-time-level multiply-upstream semi-Lagrangian schemes were examined in the case of the self-advecting, one-dimensional nonlinear momentum conservation equation. The shock formation process was analyzed. It is pointed out that the shocks cannot be created in the truncated systems satisfying the Pudykiewicz, Benoit and Staniforth criterion. The numerical integrations were restricted to 12 h. It was shown that, at least in the sub-CFL range, increased complexity of the scheme can compensate reduced horizontal resolution. A considerable sensitivity of the schemes with respect to the time step was detected. In the super-CFL mode, several windows on various time scales were found within which the Pudykiewicz, Benoit and Staniforth criterion was satisfied. The time step of 1.44 times the maximum time step allowed by the CFL criterion was used in the semi-Lagrangian runs. The super-CFL, semi-Lagrangian solutions were diverging progressively from the sub-CFL ones as the forecasts advanced. This was also reflected in the energy spectra. Unacceptably large energy losses were encountered in the super-CFL, semi-Lagrangian runs. Most of these losses could be explained by the reduced mean wind speed, i.e., the amplitude of the zero wavenumber wave. At the same time, the energy content in the shorter waves increased. In a more complex model, such a situation would resemble a loss of “zonal”, and an increase of “transient eddy” kinetic energy. A trajectory error measure was defined as the maximum absolute value of the distance between the actual arriving point of the particle originating at the estimated departure point, and the grid point assumed to be the arrival point in the semi-Lagrangian procedure. In contrast to the sub-CFL regime, this measure could reach a considerable fraction of the grid distance in the computations with the super-CFL time steps. In the physical system considered, the trajectories are determined only by the velocities at the departure points. With the semi-Lagrangian schemes the distances traveled by the particles are estimated on the basis of the velocities at the points downstream with respect to the departure points. Thus, unless the solution is smooth (in space and time) on the scales of the extrapolation distances/times, the upstream extrapolation does not promise the convergence of the solution.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01029598
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