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

Summary form only given, as follows. A self consistent and sufficiently accurate picture of plasma dynamics emerges from the solution of the coupled set of Maxwell-Vlasov equations. To remove the unobserved velocity space degrees of freedom the coupled moments of the Vlasov equation are calculated and truncated following certain prescribed rules. This process results in a fluid dynamics description of plasma interactions. In this paper a technique for solving these equations using a modification of the collocation method with Chebyshev polynomials is presented. An outline of the technique follows. The computational region under consideration is broken into subregions in which all functions are represented as a sum of a small number of Chebyshev polynomials. The PDE under consideration is solved for one time step in each subregion through Runge-Kutta integration The boundaries of the subregions are then allowed to move by analytically continuing the solution from one region into another. The problem is solved again in each new subregion and the process is continued for as many time steps as needed. The problem of the Gibb´s Phenomenon is avoided by using a spline to connect a region on one side of a singularity with the other side. The method is applied to four types of equations: (1) the wave equation; (2) the Burgers equation; (3) the inviscid Burgers equation with a singularity; (4) Laplace´s equation. In all cases the numerical solution is compared to the analytic solution.