An iterative method to solve the nonlinear Poisson's equation in the case of plasma tangential discontinuities
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In order to determine the electric potential in collisionless tangential discontinuities of a magnetized plasma, it is required to solve a non-linear Poisson's equation with sources of charge and current depending on the actual potential solution. This non-linear second-order differential equation is solved by an iterative method. This leads to an ordered sequence of non-linear algebraic equations for each successive approximation of the actual electric potential. It is shown that the method holds for transitions with characteristic thicknesses (D) as thin as five Debye lengths (λ). For smaller thicknesses, when D shrinks to 3λ or less, the method fails because in that case the iteration procedure does no longer converge. Numerical results are shown for an ion-dominated layer (D ∼ 102 - 103λ), as well as for two electron-dominated layers characterized by D ≈ 5λ and D ≈ 2.5λ, respectively. In all cases considered in this paper, the relative error on the electric potential obtained as a solution of the quasi-neutrality approximation is of the order of the relative charge density. When the method holds, each successive approximation reduces the relative error on the potential by roughly a factor of 10. For space plasma boundary layers, the quasi-neutrality approximation can be used with much confidence since their thickness is always much larger than the local Debye length.
CitationRoth, M.; Lemaire, J.; Misson, A. (1990). An iterative method to solve the nonlinear Poisson's equation in the case of plasma tangential discontinuities. , Journal of Computational Physics, Vol. 86, Issue 2, 466-486, DOI: 10.1016/0021-9991(90)90109-E.