Application of the Chebyshev collocation method to the types ofpartial differential equations which occur in plasma physics

摘要

Summary form only given, as follows. A self consistent andsufficiently accurate picture of plasma dynamics emerges from thesolution of the coupled set of Maxwell-Vlasov equations. To remove theunobserved velocity space degrees of freedom the coupled moments of theVlasov equation are calculated and truncated following certainprescribed rules. This process results in a fluid dynamics descriptionof plasma interactions. In this paper a technique for solving theseequations using a modification of the collocation method with Chebyshevpolynomials is presented. An outline of the technique follows. Thecomputational region under consideration is broken into subregions inwhich all functions are represented as a sum of a small number ofChebyshev polynomials. The PDE under consideration is solved for onetime step in each subregion through Runge-Kutta integration Theboundaries of the subregions are then allowed to move by analyticallycontinuing the solution from one region into another. The problem issolved again in each new subregion and the process is continued for asmany time steps as needed. The problem of the Gibb's Phenomenon isavoided by using a spline to connect a region on one side of asingularity with the other side. The method is applied to four types ofequations: (1) the wave equation; (2) the Burgers equation; (3) theinviscid Burgers equation with a singularity; (4) Laplace's equation. Inall cases the numerical solution is compared to the analytic solution