A nodal integral method for the Fokker-Planck collision operator

Thesis/Dissertation · Wed Jan 01 04:00:00 EST 1992

OSTI ID:7161772

McArdle, K R

A nodal integral method is developed for the solution of the steady-state Fokker-Planck collision operator plus source under the assumption of spatial homogeneity, velocity space azimuthal symmetry and a spherically symmetric background distribution. The formalism of the method involves transverse averaging the equation over a computational element or node and exactly solving the resulting ordinary differential equations (ODEs) in the remaining independent variable. Approximations must be made to source-like terms introduced through the transverse averaging procedure by expanding them in Legendre polynomials and truncating at the first term (the constant term). The Fokker-Planck equation is a PDE with variable coefficients, which is written in spherical velocity coordinates for many applications ([upsilon] and [mu] = cos [theta]). The variable coefficients introduce weight functions into the transverse averaged variable which must be approximated to relate them to simple averages of the solution over the edges of a node. For an elementary test problem with Maxwellian solution the nodal integral solution is exact. In other problems, results compared with a finite difference method demonstrate that the nodal integral method is: more efficient for some problems; has superior solutions; and has very accurate conservation of particle and energy balance. In a test problem which represents electrostatic trapping of charged particles, a Maxwellian peak at [upsilon] = 0 is computed more accurately for the nodal method for smaller N[sub [mu]] (number of meshes in [mu]) and a much smaller N[sub [upsilon]] (number of meshes in [upsilon]). The nodal integral method has a computational advantage over the finite difference due to its exact Maxwellian solution, the corresponding finite difference method had difficulty computing the Maxwellian peak accurately unless many [upsilon] meshes are used.

• Research Organization: Virginia Univ., Charlottesville, VA (United States)
• OSTI ID: 7161772
• Country of Publication: United States
• Language: English

Related Subjects

661300* -- Other Aspects of Physical Science-- (1992-)
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
CALCULATION METHODS
DIFFERENTIAL EQUATIONS
EQUATIONS
FINITE DIFFERENCE METHOD
FOKKER-PLANCK EQUATION
ITERATIVE METHODS
MESH GENERATION
NODAL EXPANSION METHOD
NUMERICAL SOLUTION
PARTIAL DIFFERENTIAL EQUATIONS