Suppression of recurrence in the Hermite-spectral method for transport equations. (English) Zbl 1406.35404
The authors consider the equation \(\frac{\partial f}{\partial t}+\xi \frac{\partial f}{\partial x} + E(x,\xi) \frac{\partial f}{\partial \xi}=0\), \(t>0\), \(x\), \(\xi \in \mathbb R\) in a class of periodic functions in \(x\). A given function \(E(x,\xi)\) is also periodic. The recurrence is an unphysical periodic behavior in the numerical solutions of the equation. A filter is constructed by using the Hermite-spectral method. It is proven and numerically validated that all the nonconstant modes are damped exponentially by the filter and therefore the recurrence is suppressed.
Reviewer: Vladimir Mityushev (Kraków)
MSC:
| 35Q83 | Vlasov equations |
| 76M22 | Spectral methods applied to problems in fluid mechanics |
| 93E11 | Filtering in stochastic control theory |
Software:
VadorReferences:
| [1] | H. Abbasi, M. H. Jenab, and H. H. Pajouh, Preventing the recurrence effect in the Vlasov simulation by randomizing phase-point velocities in phase space, Phys. Rev. E., 84 (2011), 036702. |
| [2] | R. Balesçu, Statistical Mechanics of Charged Particles, Wiley Interscience, New York, 1963. · Zbl 0125.23106 |
| [3] | C. K. Birdsall and A. B. Langdon, Plasma Physics via Computer Simulation, Institute of Physics Publishing, Bristol, UK, 1991. |
| [4] | Z. Cai, R. Li, and Y. Wang, Solving Vlasov equations using NR\(xx\) method, SIAM J. Sci. Comput., 35 (2013), pp. A2807–A2831. · Zbl 1287.35090 |
| [5] | E. Camporeale, G. L. Delzanno, B. K. Bergen, and J. D. Moulton, On the velocity space discretization for the Vlasov-Poisson system: Comparison between implicit Hermite spectral and particle-in-cell methods, Commun. Comput. Phys., 198 (2016), pp. 47–58. · Zbl 1344.35147 |
| [6] | C. Z. Cheng and G. Knorr, The integration of the Vlasov equation in configuration space, J. Comput. Phys., 22 (1976), pp. 330–351. |
| [7] | S. Dawson and M. F. Wheeler, Particle simulation of plasma, Rev. Modern Phys., 55 (1983), pp. 403–447. |
| [8] | L. Einkemmer and A. Ostermann, A strategy to suppress recurrence in grid-based Vlasov solvers, Eur. Phys. J. D., 68 (2014), 197. |
| [9] | L. Einkemmer and A. Ostermann, Convergence analysis of a discontinuous Galerkin/Strang splitting approximation for the Vlasov–Poisson equations, SIAM J. Numer. Anal., 52 (2014), pp. 757–778. · Zbl 1302.82108 |
| [10] | F. Filbet and E. Sonnendrücker, Comparison of Eulerian Vlasov solvers, Comput. Phys. Commun., 150 (2003), pp. 247–266. · Zbl 1196.82108 |
| [11] | F. Filbet, E. Sonnendrücker, and P. Bertrand, Conservative numerical schemes for the Vlasov equation, J. Comput. Phys., 172 (2001), pp. 166–187. · Zbl 0998.65138 |
| [12] | R. J. Goldston and P. H. Rutherford, Introduction to Plasma Physics, CRC Press, Boca Raton, FL, 2016. · Zbl 1045.82001 |
| [13] | D. Gottlieb and J. S. Hesthaven, Spectral methods for hyperbolic problems, J. Comput. Appl. Math., 128 (2001), pp. 83–131. · Zbl 0974.65093 |
| [14] | F. C. Grant and M. R. Feix, Fourier-Hermite solutions of the Vlasov equations in the linearized limit, Phys. Fluids, 10 (1967), pp. 696–702. |
| [15] | R. W. Hamming, Introduction to Applied Numerical Analysis, Taylor & Francis, Philadelphia, 1989. · Zbl 0705.65002 |
| [16] | R. E. Heath, I. M. Gamba, P. J. Morrison, and C. Michler, A discontinuous Galerkin method for the Vlasov-Poisson system, J. Comput. Phys., 231 (2012), pp. 1140–1174. · Zbl 1244.82081 |
| [17] | P. P. Hilscher, K. Imadera, J. Q. Li, and Y. Kishimoto, The effect of weak collisionality on damped modes and its contribution to linear mode coupling in gyrokinetic simulation, Phys. Plasmas, 20 (2013), 082127. |
| [18] | P. Hilton and J. Pedersen, The Ballot problem and Catalan numbers, Nieuw Arch. Wiskd., 8 (1990), pp. 209–216. · Zbl 0714.05002 |
| [19] | R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles, McGraw-Hill, New York, 1981. · Zbl 0662.76002 |
| [20] | J. P. Holloway, Spectral velocity discretizations for the Vlasov-Maxwell equations, Transp. Theor. Stat., 25 (1996), pp. 1–32. · Zbl 0864.76101 |
| [21] | T. Hou and R. Li, Computing nearly singular solutions using pseudo-spectral methods, J. Comput. Phys., 226 (2007), pp. 379–397. · Zbl 1310.76127 |
| [22] | B. K\aagström, Bounds and perturbation bounds for the matrix exponential, BIT, 17 (1977), pp. 39–57. · Zbl 0356.65034 |
| [23] | L. Landau, On the vibrations of the electronic plasma, Zh. Eksp. Teor. Fiz., 10 (1946), 25. · Zbl 0063.03439 |
| [24] | T. Nakamura and T. Yabe, Cubic interpolated propagation scheme for solving the hyper-dimensional Vlasov-Poisson equation in phase space, Comput. Phys. Commun, 120 (1999), pp. 122–154. · Zbl 1001.82003 |
| [25] | J. T. Parker and P. J. Dellar, Fourier-Hermite spectral representation for the Vlasov-Poisson system in the weakly collisional limit, J. Plasma. Phys., 81 (2015), 305810203. |
| [26] | O. Pezzi, E. Camproeale, and F. Valentini, Collisional effects on the numerical recurrence in Vlasov-Poisson simulation, Phys. Plasmas, 23 (2016), 022103. |
| [27] | E. Pohn, M. Shoucri, and G. Kamelander, Eulerian Vlasov codes, Comput. Phys. Commun., 166 (2005), pp. 81–93. · Zbl 1196.82113 |
| [28] | J. Qiu and C. Shu, Positivity preserving semi-Lagrangian discontinuous Galerkin formulation: Theoretical analysis and application to the Vlasov-Poisson system, J. Comput. Phys., 230 (2011), pp. 8386–8409. · Zbl 1273.65147 |
| [29] | P. P. J. M. Schram, Kinetic Theory of Gases and Plasmas, Springer, New York, 1991. |
| [30] | E. Sonnendrücker, Approximation numérique des équations de Vlasov-Maxwell, Notes du cours de M2, Université de Strasbourg, 2010. |
| [31] | E. Sonnendrücker, J. Roche, P. Betrand, and A. Ghizzo, The semi-Lagrangian method for the numerical resolution of Vlasov equations, J. Comput. Phys., 149 (1998), pp. 201–220. · Zbl 0934.76073 |
| [32] | V. Vedenyapin and A. Sinitsyn, Kinetic Boltzmann, Vlasov and Related Equations, Elsevier, Amsterdam, 2011. · Zbl 1230.35004 |
| [33] | J. H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, UK, 1965. · Zbl 0258.65037 |
| [34] | T. Zhou, Y. Guo, and C. W. Shu, Numerical study on Landau damping, Phys. D, 157 (2001), pp. 322–333. · Zbl 0972.82083 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
