Spline Function Expression by Orthogonal Polynomial Weighted Sampling for High-Speed Electric Field Calculation

Abstract

In the simulation of plasma processes, minimizing the error associated with the electric field calculation in the vicinity of the sheath is an important consideration. The sheath length scale when compared to the plasma size and fact that the electric fields must be solved for self-consistently with equations describing the plasma chemistry make the electric field solution particularly expensive. Because an internal electric field is calculated based on the density of charged particles, it is necessary to obtain the density of charged particles correctly but at minimum possible expense. We describe an electric field simulation with high speed and good accuracy enabled by the description of the density of charges by spline functions. Preciously, we have taken advantage of the orthogonality of the Legendre polynomial in our methodology described as LPWS (Legendre Polynomial Weighted Sampling). Sampling with other orthogonal functions is also possible and as such we have generalized our method. The generalization, Orthogonal Polynomial Weighted Sampling (OPWS), whereby the method of obtaining the coefficient of B-spline function from the coefficient of the orthogonal expansion has been developed and is described in this paper.