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calls | CPU (sec) | ||||
400 | 400 | 160k | 480k | 20 | 358k | 256 |
100 | 400 | 160k | 480k | 14 | 273k | 218 |
400 | 100 | 160k | 480k | 15 | 301k | 229 |
400 | 400 | 40k | 480k | 7 | 174k | 151 |
400 | 400 | 160k | 960k | 9 | 402k | 345 |
Stochastic particle methods for the numerical treatment of the Wigner equation are considered. The approximation properties of these methods depend on several numerical parameters. Such parameters are the number of particles, a time step (if transport and other processes are treated separately) and the grid size (used for the discretization of the position and the wave-vector). A stochastic algorithm without time discretization error is introduced. Its derivation is based on the theory of piecewise deterministic Markov processes. Numerical experiments are performed in a one-dimensional test case. Approximation properties with respect to the grid size and the number of particles are studied. Convergence of a time-splitting scheme to the no-splitting algorithm is demonstrated. The no-splitting algorithm is shown to be more efficient in terms of computational effort.
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Table 1.
Properties of the algorithm for various sets of cancellation parameters. The quantity "calls" denotes the number of calls to the cancellation procedure and
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calls | CPU (sec) | ||||
400 | 400 | 160k | 480k | 20 | 358k | 256 |
100 | 400 | 160k | 480k | 14 | 273k | 218 |
400 | 100 | 160k | 480k | 15 | 301k | 229 |
400 | 400 | 40k | 480k | 7 | 174k | 151 |
400 | 400 | 160k | 960k | 9 | 402k | 345 |
Table 2. Properties of the time-splitting algorithm and the no-splitting algorithm. The quantities "err-max" and "err-aver" denote, respectively, the maximum and the average (over the cells) of the absolute differences between the measured density and the reference solution. The last column provides the measurements of the first, second and third cancellation time
CPU (sec.) | err-max | err-aver | canc. times | |
1 | 278 | 0.0106 | 0.0024 | 3.0000, 6.0000, 8.0000 |
0.4 | 395 | 0.0053 | 0.0009 | 2.4000, 4.4000, 6.4000 |
0.1 | 928 | 0.0019 | 0.0004 | 2.0000, 3.8000, 5.4870 |
0.05 | 1628 | 0.0018 | 0.0003 | 1.9500, 3.6700, 5.2730 |
0.025 | 3028 | 0.0017 | 0.0003 | 1.9222, 3.6222, 5.2182 |
no-splitting | 256 | 0.0013 | 0.0002 | 1.8927, 3.5716, 5.1428 |
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