Discretization and Solution of the Poisson Equations

Before we discuss in detail the numerical discretization scheme used for the Poisson equation, which by the way is very similar to the discretization of the radial Schrodinger equation given in Eq. (9.120) [491], we sum up some of its general features. We consider the general form of the radial Poisson equation, 

A numerical treatment requires an approximation for the second derivative. Application of a finite difference method transforms the second derivative at a grid point Sj. into a linear combination of function values ym at contiguous grid points Sm around the point Sfc. The differential equation at the grid point Sfc therefore becomes a linear difference equation in the unknown function values ym at successive grid points. The resulting set of linear equations can be combined into an n x n) matrix equation, which is inhomogeneous in this case. 

The elements of the vector y = y, . y, . y Y are the function values to be determined. The elements of the band matrix A are determined by the values Ffc of the coefficient function F(s) and by the finite difference approximation to the second derivative. The elements of the inhomogeneity vector g = ( 1, , gk, , gn) depend analogously on the values of the inhomogeneity function G(s). After the setup of the required matrices and of the 

The discretization scheme, which leads to an error 0 h ) for second-order differential equations (without first derivative) with the lowest number of points in the difference equation, is the method frequently attributed to Nu-merov [494,499]. It can be efficiently employed for the transformed Poisson Eq. (9.232). In this approach, the second derivative at grid point Sjt is approximated by the second central finite difference at this point, corrected to order h, and requires values at three contiguous points (see appendix G for details). Finally, we obtain tri-diagonal band matrix representations for both the second derivative and the coefficient function of the differential equation. The resulting matrix A and the inhomogeneity vector g are then 

In some cases, where iXi + iXj — 1 0 so that Go oo, it is not possible to correct the Numerov scheme for the Poisson equation [491,492]. In these cases one can use a five-point central difference formula [498] (also with truncation error of order h ) which leaves the coefficient function on the diagonal. However, this five-point formula leads to difficulties at the boundaries, where values at grid points s i (at the lower boundary) and s +2 (at the upper boundary) would be needed, which lie outside the range of definition. 

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