## An iterative method to solve the nonlinear Poisson equation in the case of plasma tangential discontinuities.(English)Zbl 0682.76101

Summary: In order to determine the electric potential in collisionless tangential discontinuities of a magnetized plasma, it is required to solve a nonlinear Poisson’s equation with sources of charge and current depending on the actual potential solution. This nonlinear second-order differential equation is solved by an iterative method. This leads to an ordered sequence of nonlinear 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 ($$\lambda)$$. For smaller thicknesses, when D shrinks to $$3\lambda$$ 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 $$(D\sim 10^ 2-10^ 3\lambda)$$, as well as for two electron-dominated layers characterized by $$D\approx 5\lambda$$ and $$D\approx 2.5\lambda$$, 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.

### MSC:

 76X05 Ionized gas flow in electromagnetic fields; plasmic flow 76M99 Basic methods in fluid mechanics

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### References:

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