Highly efficient numerical algorithm based on random trees for accelerating parallel Vlasov-Poisson simulations

Rodriguez Rozas, Angel ; Juan A., Acebrón

HAL, hal-00920998 / Harvested from HAL

Résumé

An efficient numerical method based on a probabilistic representation for the Vlasov-Poisson system of equations in the Fourier space has been derived. This has been done theoretically for arbitrary dimensional problems, and particularized to unidimensional problems for numerical purposes. Such a representation has been validated theoretically in the linear regime comparing the solution obtained with the classical results of the linear Landau damping. The numerical strategy followed requires generating suitable random trees combined with a Padé approximant for approximating accurately a given divergent series. Such series are obtained by summing the partial contributions to the solution coming from trees with arbitrary number of branches. These contributions, coming in general from multi-dimensional definite integrals, are efficiently computed by a quasi-Monte Carlo method. It is shown how the accuracy of the method can be effectively increased by considering more terms of the series. The new representation was used successfully to develop a Probabilistic Domain Decomposition method suited for massively parallel computers, which improves the scalability found in classical methods. Finally, a few numerical examples based on classical phenomena such as the non-linear Landau damping, and the two streaming instability are given, illustrating the remarkable performance of the algorithm, when compared the results with those obtained using a classical method.

Détail

Publié le : 2013-10-01
Classification:  Probabilistic Domain Decomposition; Probabilistic representation of solution of partial differential equations; Quasi-Monte carlo,  ACM: G.: Mathematics of Computing/G.4: MATHEMATICAL SOFTWARE/G.4.0: Algorithm design and analysis,  ACM: G.: Mathematics of Computing/G.4: MATHEMATICAL SOFTWARE/G.4.3: Efficiency,  ACM: G.: Mathematics of Computing/G.1: NUMERICAL ANALYSIS/G.1.1: Interpolation,  ACM: G.: Mathematics of Computing/G.1: NUMERICAL ANALYSIS/G.1.8: Partial Differential Equations/G.1.8.0: Domain decomposition methods,  ACM: G.: Mathematics of Computing/G.1: NUMERICAL ANALYSIS/G.1.8: Partial Differential Equations/G.1.8.5: Hyperbolic equations,  ACM: G.: Mathematics of Computing/G.1: NUMERICAL ANALYSIS/G.1.8: Partial Differential Equations,  ACM: G.: Mathematics of Computing/G.1: NUMERICAL ANALYSIS/G.1.1: Interpolation/G.1.1.4: Spline and piecewise polynomial interpolation,  ACM: D.: Software/D.1: PROGRAMMING TECHNIQUES/D.1.3: Concurrent Programming/D.1.3.1: Parallel programming,  ACM: D.: Software/D.2: SOFTWARE ENGINEERING/D.2.4: Software/Program Verification,  ACM: D.: Software/D.2: SOFTWARE ENGINEERING/D.2.4: Software/Program Verification/D.2.4.7: Statistical methods,  ACM: J.: Computer Applications,  ACM: J.: Computer Applications/J.2: PHYSICAL SCIENCES AND ENGINEERING/J.2.8: Physics,  ACM: G.: Mathematics of Computing/G.4: MATHEMATICAL SOFTWARE/G.4.4: Parallel and vector implementations,  ACM: G.: Mathematics of Computing/G.4: MATHEMATICAL SOFTWARE/G.4.6: Reliability and robustness,  ACM: G.: Mathematics of Computing/G.3: PROBABILITY AND STATISTICS,  ACM: G.: Mathematics of Computing/G.3: PROBABILITY AND STATISTICS/G.3.7: Probabilistic algorithms (including Monte Carlo),  ACM: G.: Mathematics of Computing/G.3: PROBABILITY AND STATISTICS/G.3.15: Stochastic processes,  ACM: G.: Mathematics of Computing/G.2: DISCRETE MATHEMATICS/G.2.1: Combinatorics,  ACM: G.: Mathematics of Computing/G.2: DISCRETE MATHEMATICS/G.2.1: Combinatorics/G.2.1.0: Combinatorial algorithms,  ACM: G.: Mathematics of Computing/G.1: NUMERICAL ANALYSIS,  [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation,  [SPI.PLASMA]Engineering Sciences [physics]/Plasmas,  [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP],  [MATH.MATH-PR]Mathematics [math]/Probability [math.PR],  [INFO.INFO-NA]Computer Science [cs]/Numerical Analysis [cs.NA],  [INFO.INFO-DC]Computer Science [cs]/Distributed, Parallel, and Cluster Computing [cs.DC],  [PHYS.PHYS.PHYS-COMP-PH]Physics [physics]/Physics [physics]/Computational Physics [physics.comp-ph]