A deterministic particle method for the Vlasov–Fokker–Planck equation in one dimension

• Published in: Journal of computational and applied mathematics 2008-04, Vol.213 (2), p.316-365
• Main Authors: Wollman, Stephen, Ozizmir, Ercument
• Format: Article
• Language: English
• Subjects: Collisional plasma; Deterministic particle method; Exact sciences and technology; Functional analysis; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical approximation; Numerical methods in probability and statistics; Partial differential equations; Sciences and techniques of general use; Vlasov–Fokker–Planck equation
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2007.01.008

The Vlasov–Fokker–Planck equation is a model for a collisional, electrostatic plasma. The approximation of this equation in one spatial dimension is studied. The equation under consideration is linear in that the electric field is given as a known function that is not internally consistent with the phase space distribution function. The approximation method applied is the deterministic particle method described in Wollman and Ozizmir [Numerical approximation of the Vlasov–Poisson–Fokker–Planck system in one dimension, J. Comput. Phys. 202 (2005) 602–644]. For the present linear problem an analysis of the stability and convergence of the numerical method is carried out. In addition, computations are done that verify the convergence of the numerical solution. It is also shown that the long term asymptotics of the computed solution is in agreement with the steady state solution derived in Bouchut and Dolbeault [On long time asymptotics of the Vlasov–Fokker–Planck equation and of the Vlasov–Poisson–Fokker–Planck system with coulombic and Newtonian potentials, Differential Integral Equations 8(3) (1995) 487–514].