Improvements of finite differences methods

Details Improvements of finite differences methods Improvements of finite differences methods

G - Physics

06

F

G06F 17/13 (2006.01)

Patent

CA 2639730

Many important physical phenomena are described by partial differential equations (PDEs) of a function f(x,y,z) constrained by boundary conditions. Solutions of such 'boundary value problems' are often calculated numerically using so-called 'finite differences' (FD) methods. Essentially, a regular array, or 'grid', of points is established in the spatial domain, and an array of difference equations are solved numerically to yield values of f(x,y,z) that are consistent with the boundary conditions at the locations of the grid points. However, the boundary conditions are often not initially specified precisely at the locations of grid points, in particular, when such boundary conditions are represented by curved or slanted surfaces which do not coincide precisely with grid point array. Typically, the locations of such boundary surfaces are approximated by grid points located proximal to the boundary surfaces, while the values of f(x,y,z) at these 'boundary grid points' are, nevertheless, assigned the values of the boundary surfaces proximal to the 'boundary grid points'. Such approximations of the original boundary conditions lead to errors in the subsequent determination of the function f(x,y,z) by finite differences methods. The present invention provides methods by which the values assigned to the boundary grid points are not the values of f(x,y,z) of non-coincident boundary surfaces proximal to the boundary grid points, but rather, values are assigned to the boundary grid points that more accurately account for the differences in locations between each original boundary surface and that of each respective proximal boundary grid point. Consequently, the subject invention provides methods by which the function f(x,y,z) may be calculated with FD methods with greater accuracy than methods. The invention is described herein particularly with regard to solving Laplace's equation to determine potential distributions in electron and/or ion optical devices.

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