The singularity of the two-fluid plasma equations, its relations to boundary conditions, and the numerical solution of these equations

Abstract

The equations describing the two-fluid model of a plasma contain a removable singularity at the ion sound velocity. Taking into account a non-zero ion temperature, the irregular point is located between the centre of the plasma and the wall. It is advisable to treat the inner interval between the centre and the irregular point and the outer one between this point and the wall separately. Taylor series yield a smooth solution through this point. Readily manageable numerical solution methods are stable in the outer interval but very unstable in the inner one. The domain of the parameters is determined in which the one-fluid model results in a very useful approximation throughout the inner interval. With it, the space charge density can also be estimated well. At the irregular point, the missing accurate boundary values of the two-fluid equations required for both the intervals can be determined by means of a few steps of a shooting method starting at the results of the one-fluid model. The instability of the treated differential equations ascertained in the inner interval becomes comprehensible by using the Lyapunov criterion. The results obtained in the inner interval can be used as start values for other numerical methods to improve the results. The methods explained here allow us to obtain usable numerical results for the two-fluid model in a large interesting domain by a relatively little effort.