Numerical Algrorithm of 1D Dusty Void Mo del Formation

DAYS on DIFFRACTION 2013 1 Numerical Algrorithm of 1D Dusty Void Model Formation Kravchenko O.V. Bauman Moscow State Technical University, e-mail: olekravchenko@gmail.com Pustovoit V.I. ”Scientific and Technological Center of Unique Instrumentation” of the Russian Academy of Sciences, e-mail: vlad pst@yahoo.com In present work we consider a one–dimentional model of dusty void formation according to from numerical methods point of view. Such kind investigation is related to the problem of modelling a quasi–periodic structure of electron clusters both dusty [1] and electron–ion plasma [2] fluid. We consider the following system of governing equations in vector form:     ∂u(x, t) ∂F (u(x, t)) nd 0 + = f (x, t), u(x, t) = , f (x, t) = , (1) vd (a/(b + |vi |3 ) − 1)E − α0 vd ∂t ∂t     ∂w(x, t) −ne E/τi ne , g(x, t) = (2) = g, w(x, t) = 1 − nd − ne E ∂x in the rectangle x ∈ [0, L], t ∈ [0, T ] under the initial–boundary conditions       ∂nd (L, t) 0 α1 0 u(x, 0) = , w(0, t) = , w(L, t) = , = 0. 5 · 10−4 β1 0 ∂x (3) The algrorithm of integrating of (1)–(3) is the following: • Computation of the initial–value problem (1), (3) by one of the explicit first order schemes [3], for example Lax–Friedrichs scheme (the simpliest one) on time step of integration tn ⇒ u⋆ is known. • Computation of the boundary–value problem (2), (3) by tridiagonal block algorithm on the same time step of integration tn ⇒ w is known. • Recomputation of (1), (3) on the next time step of integration tn+1 with respect to u⋆ , w. References [1] K. Avinash, A. Bhattacharjee, and S. Hu, “Nonlinear Theory of Void Formation in Colloidal Plasmas,” Phys. Rev. Lett., 90, 075001–4 pp. (2003). [2] V.I. Pustovoit, “Mechanism of Lightning Discharge,” Journal of Communications Technology and Electronics, 51, No. 8, 937–943 pp. (2006). [3] K.M. Magomedov, A.S. Kholodov, Finite difference–characteristic method, Publishing House ”Science”, Moscow (1988) [in Russian].