A class of difference schemes with flexible local approximation

Journal Article · Fri Jan 20 00:00:00 EST 2006 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2005.06.011· OSTI ID:20687276

Tsukerman, Igor ^[1]

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1. Department of Electrical and Computer Engineering, University of Akron, Akron, OH 44325-3904 (United States)

Solutions of many physical problems have salient local features that are qualitatively known a priori (for example, singularities at point sources, edge and corners; boundary layers; derivative jumps at material interfaces; strong dipole field components near polarized spherical particles; cusps of electronic wavefunctions at the nuclei; electrostatic double layers around colloidal particles, etc.) The known methods capable of providing flexible local approximation of such features include the generalized finite element - partition of unity method, special variational-difference schemes in broken Sobolev spaces, and a few other specialized techniques. In the proposed new class of Flexible Local Approximation MEthods (FLAME), a desirable set of local approximating functions (such as cylindrical or spherical harmonics, plane waves, harmonic polynomials, etc.) defines a finite difference scheme on a chosen grid stencil. One motivation is to minimize the notorious 'staircase' effect at curved and slanted interface boundaries. However, the new approach has much broader applications. As illustrative examples, the paper presents arbitrarily high order 3-point schemes for the 1D Schroedinger equation and a 1D singular equation, schemes for electrostatic interactions of colloidal particles, electromagnetic wave propagation and scattering, plasmon resonances. Moreover, many classical finite difference schemes, including the Collatz 'Mehrstellen' schemes, are direct particular cases of FLAME.

• OSTI ID: 20687276
• Journal Information: Journal of Computational Physics, Vol. 211, Issue 2; Other Information: DOI: 10.1016/j.jcp.2005.06.011; PII: S0021-9991(05)00292-5; Copyright (c) 2005 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved; Country of input: International Atomic Energy Agency (IAEA); ISSN 0021-9991
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
BOLTZMANN EQUATION
BOUNDARY LAYERS
CUSPED GEOMETRIES
CYLINDRICAL CONFIGURATION
DIPOLES
ELECTROMAGNETIC RADIATION
FINITE DIFFERENCE METHOD
FINITE ELEMENT METHOD
PARTICLES
POINT SOURCES
POLYNOMIALS
RESONANCE
SCATTERING
SCHROEDINGER EQUATION
SINGULARITY
SPHERICAL HARMONICS
VARIATIONAL METHODS
WAVE FUNCTIONS
WAVE PROPAGATION