Discretization errors associated with reproducing kernel methods: one-dimensional domains
✍ Scribed by Thomas E. Voth; Mark A. Christon
- Publisher
- Elsevier Science
- Year
- 2001
- Tongue
- English
- Weight
- 590 KB
- Volume
- 190
- Category
- Article
- ISSN
- 0045-7825
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✦ Synopsis
The reproducing kernel particle method (RKPM) is a discretization technique for partial dierential equations that uses the method of weighted residuals, classical reproducing kernel theory and modi®ed 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 quanti®ed. 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 semidiscretizations. The model PDEs considered for this analysis include the parabolic and hyperbolic (®rst-and second-order wave) equations. Numerical diusivity for the former and phase speed for the latter 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 diusive and dispersive characteristics are observed when a consistent mass matrix formulation is used with the proper choice of re®nement parameter. In contrast, the row-sum lumped mass matrix formulation severely degraded performance. The asymptotic analysis indicates that very good rates of convergence (Ox 6 ±Ox 8 ) are possible when the consistent mass matrix formulation is used with an appropriate choice of re®nement parameter and quadrature rule.