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

Voth, T E; Christon, M A

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

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Abstract

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 matrixmore »« less

Authors:: Voth, T E; Christon, M A

Publication Date:: Mon Jan 10 00:00:00 EST 2000

Research Org.:: Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Sandia National Lab. (SNL-CA), Livermore, CA (United States)

Sponsoring Org.:: US Department of Energy (US)

OSTI Identifier:: 750192

Report Number(s):: SAND2000-0092J
TRN: AH200008%%59

DOE Contract Number:: AC04-94AL85000

Resource Type:: Journal Article

Resource Relation:: Other Information: Submitted to Computer Methods in Applied Mechanics and Engineering; PBD: 10 Jan 2000

Country of Publication:: United States

Language:: English

Subject:: 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; PARTIAL DIFFERENTIAL EQUATIONS; WEIGHTING FUNCTIONS; MESH GENERATION; PERFORMANCE; CALCULATION METHODS; ONE-DIMENSIONAL CALCULATIONS; KERNELS

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                    Voth, T E, and Christon, M A. Discretization errors associated with Reproducing Kernel Methods: One-dimensional domains.  United States: N. p., 2000. 
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                    Voth, T E, & Christon, M A. Discretization errors associated with Reproducing Kernel Methods: One-dimensional domains.  United States.

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                    Voth, T E, and Christon, M A. 2000.  
        "Discretization errors associated with Reproducing Kernel Methods: One-dimensional domains".  United States.  https://www.osti.gov/servlets/purl/750192.

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@article{osti_750192,

  title        = {Discretization errors associated with Reproducing Kernel Methods: One-dimensional domains},

  author       = {Voth, T E and Christon, M A},

  abstractNote = {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.},

  doi          = {},

  url          = {https://www.osti.gov/biblio/750192},
  journal      = {},
number       = ,

  volume       = ,

  place        = {United States},

  year         = {Mon Jan 10 00:00:00 EST 2000},

  month        = {Mon Jan 10 00:00:00 EST 2000}

}

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