Discretization errors associated with Reproducing Kernel Methods: One-dimensional domains

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The Reproducing Kernel Particle Method (RKPM) is a discretization technique for partial differential equations that uses the method of weighted residuals, classical reproducing kernel theory and modified kernels to produce either ``mesh-free'' or ``mesh-full'' methods. Although RKPM has many appealing attributes, the method is new, and its numerical performance is just beginning to be quantified. In order to address the numerical performance of RKPM, von Neumann analysis is performed for semi-discretizations of three model one-dimensional PDEs. The von Neumann analyses results are used to examine the global and asymptotic behavior of the semi-discretizations. The model PDEs considered for this analysis ... continued below

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29 p.

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Voth, T.E. & Christon, M.A. January 10, 2000.

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  • Sandia National Laboratories
    Publisher Info: Sandia National Labs., Albuquerque, NM, and Livermore, CA (United States)
    Place of Publication: Albuquerque, New Mexico

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The Reproducing Kernel Particle Method (RKPM) is a discretization technique for partial differential equations that uses the method of weighted residuals, classical reproducing kernel theory and modified kernels to produce either ``mesh-free'' or ``mesh-full'' methods. Although RKPM has many appealing attributes, the method is new, and its numerical performance is just beginning to be quantified. In order to address the numerical performance of RKPM, von Neumann analysis is performed for semi-discretizations of three model one-dimensional PDEs. The von Neumann analyses results are used to examine the global and asymptotic behavior of the semi-discretizations. The model PDEs considered for this analysis include the parabolic and hyperbolic (first and second-order wave) equations. Numerical diffusivity for the former and phase speed for the later are presented over the range of discrete wavenumbers and in an asymptotic sense as the particle spacing tends to zero. Group speed is also presented for the hyperbolic problems. Excellent diffusive and dispersive characteristics are observed when a consistent mass matrix formulation is used with the proper choice of refinement parameter. In contrast, the row-sum lumped mass matrix formulation severely degraded performance. The asymptotic analysis indicates that very good rates of convergence are possible when the consistent mass matrix formulation is used with an appropriate choice of refinement parameter.

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29 p.

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OSTI as DE00750192

Medium: P; Size: 29 pages

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  • Other Information: Submitted to Computer Methods in Applied Mechanics and Engineering

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  • Report No.: SAND2000-0092J
  • Grant Number: AC04-94AL85000
  • Office of Scientific & Technical Information Report Number: 750192
  • Archival Resource Key: ark:/67531/metadc708497

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  • January 10, 2000

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  • Sept. 12, 2015, 6:31 a.m.

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  • April 10, 2017, 7:07 p.m.

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Voth, T.E. & Christon, M.A. Discretization errors associated with Reproducing Kernel Methods: One-dimensional domains, article, January 10, 2000; Albuquerque, New Mexico. (digital.library.unt.edu/ark:/67531/metadc708497/: accessed October 22, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.