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A class of difference schemes with flexible local approximation. (English) Zbl 1082.65108

The focus of this paper is on “flexible local approximation” and on methods capable of proving it, employing a variety of approximating functions not at all limited to polynomials. New ideas are considered. A new class of flexible local approximation methods (FLAME) is introduced, where the difference scheme is defined by the chosen set of local basis functions and the grid stencil. The motivation of developing this class of methods is to minimize the notorious staircase effect at curved and slanted interface boundaries on regular Cartesian grids.
As illustrative examples, the paper presents arbitrarily high order 3-point schemes for the 1D Schrödinger equation and a 1D singular equation, schemes for electrostatic interactions of colloidal particles, electromagnetic wave propagation and scattering, and plasmon resonances.

MSC:

65N06 Finite difference methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
78M20 Finite difference methods applied to problems in optics and electromagnetic theory
78A40 Waves and radiation in optics and electromagnetic theory
78A45 Diffraction, scattering
82C80 Numerical methods of time-dependent statistical mechanics (MSC2010)

Software:

APBS; FLAME; Mfree2D
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Full Text: DOI

References:

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