Discretization Errors Associated with Reproducing Kernel Methods: One-Dimensional Domains

The Reproducing Kernel Particle Method (RKPM) is a discretization technique for partial di erential equations that uses the method of weighted residuals to produce either \mesh-free" or \mesh-full" methods. The technique employs classical reproducing kernel theory and incorporates modi ed kernels to account for inconsistencies arising from the (unavoidable) introduction of discrete integration of the reproducing equation and nite domains. The use of a variable kernel function width allows for resolution of multiple \scales" of the solution providing a framework for hierarchical simulations. Although RKPM has many appealing attributes, the method is new, and its numerical performance is just beginning to be quanti ed with respect to more traditional discretization techniques. In order to address the numerical performance of RKPM, von Neumann analysis is used to evaluate the spectral behavior of the resulting semi-discretizations. Detailed studies of the e ect of kernel width, integration rule and mass matrix formulation are investigated. For generality, the studies are conducted on a set of one-dimensional model partial di erential equations. Results of the von Neumann analyses are used to evaluate the global behavior of the semi-discretizations, i.e. behavior over the entire range of resolvable wavelengths (accuracy-in-the-large). In addition, the results are used to infer the asymptotic behavior (or rate of convergence) of the semi-discretizations as the particle spacing goes to zero (accuracy-in-the-small; [26]). It has been suggested that, in practice, accuracy-in-the-large is a better indicator of a method's accuracy than are the \pristine" convergence rates suggested by the accuracy-in-the-small analysis [7, 26]. The model partial di erential equations considered for this analysis are the one-dimensional parabolic and hyperbolic ( rst and second-order wave) equations. E ective (numerical) di usivity 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 di usive (parabolic) and dispersive (hyperbolic) characteristics are observed for the consistent mass matrix with the proper choice of re nement parameter. In contrast, the row-sum lumped mass matrix formulation severely under-di uses short wavelength waves for the parabolic equations and results in signi cantly lagging phase speed for the hyperbolic equations. The \higher-order" mass matrix formulation results in only a small improvement in the di usive and dispersive characteristics relative to the lumped mass matrix. The asymptotic analysis indicates that very good rates of convergence are possible for both parabolic and hyperbolic problems provided that the consistent mass matrix formulation is used with the appropriate choice of re nement parameter. Deviations from this re nement parameter, or use of lumped or higher order mass matrices, modify the convergence rates making them comparable to those for linear FEM.