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Collocation Methods (collocation + methods)

Distribution by Scientific Domains

Mathematics and Statistics	50%
Engineering	50%

Selected Abstracts

Effective condition number of Trefftz methods for biharmonic equations with crack singularities

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 2 2009
Zi-Cai Li
Abstract The paper presents the new stability analysis for the collocation Trefftz method (CTM) for biharmonic equations, based on the new effective condition number Cond_eff. The Trefftz method is a special spectral method with the particular solutions as admissible functions, and it has been widely used in engineering. Three crack models in Li et al. (Eng. Anal. Boundary Elements 2004; 28:79,96; Trefftz and Collocation Methods. WIT Publishers: Southampton, Boston, 2008) are considered, and the bounds of Cond_eff and the traditional condition number Cond are derived, to give the polynomial and the exponential growth rates, respectively. The stability analysis explains well the numerical experiments. Hence, the new Cond_eff is more advantageous than Cond. Besides since the bounds of Cond_eff and Cond involve the estimation of the minimal singular value ,min of the discrete matrix F, and since the estimation of ,min is challenging and difficult, the proof for lower bounds of ,min in this paper is important and intriguing. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Collocation methods based on radial basis functions for solving stochastic Poisson problems

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 3 2007
Somchart ChantasiriwanArticle first published online: 19 JUN 200
Abstract Collocation methods based on radial basis functions can be used to provide accurate solutions to deterministic problems. For stochastic problems, accurate solutions may not be desirable if they are too sensitive to random inputs. In this paper, four methods are used to solve stochastic Poisson problems by expressing solutions in terms of source terms and boundary conditions. Comparison among the methods reveals that the method based on fundamental solutions performs better than other methods. Copyright © 2006 John Wiley & Sons, Ltd. [source]

Radial point interpolation based finite difference method for mechanics problems

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 7 2006
G. R. Liu
Abstract A radial point interpolation based finite difference method (RFDM) is proposed in this paper. In this novel method, radial point interpolation using local irregular nodes is used together with the conventional finite difference procedure to achieve both the adaptivity to irregular domain and the stability in the solution that is often encountered in the collocation methods. A least-square technique is adopted, which leads to a system matrix with good properties such as symmetry and positive definiteness. Several numerical examples are presented to demonstrate the accuracy and stability of the RFDM for problems with complex shapes and regular and extremely irregular nodes. The results are examined in detail in comparison with other numerical approaches such as the radial point collocation method that uses local nodes, conventional finite difference and finite element methods. Copyright © 2006 John Wiley & Sons, Ltd. [source]

Stable high-order finite-difference methods based on non-uniform grid point distributions

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2008
Miguel Hermanns
Abstract It is well known that high-order finite-difference methods may become unstable due to the presence of boundaries and the imposition of boundary conditions. For uniform grids, Gustafsson, Kreiss, and Sundström theory and the summation-by-parts method provide sufficient conditions for stability. For non-uniform grids, clustering of nodes close to the boundaries improves the stability of the resulting finite-difference operator. Several heuristic explanations exist for the goodness of the clustering, and attempts have been made to link it to the Runge phenomenon present in polynomial interpolations of high degree. By following the philosophy behind the Chebyshev polynomials, a non-uniform grid for piecewise polynomial interpolations of degree q,N is introduced in this paper, where N + 1 is the total number of grid nodes. It is shown that when q=N, this polynomial interpolation coincides with the Chebyshev interpolation, and the resulting finite-difference schemes are equivalent to Chebyshev collocation methods. Finally, test cases are run showing how stability and correct transient behaviours are achieved for any degree q[source]

A numerical study of the accuracy and stability of symmetric and asymmetric RBF collocation methods for hyperbolic PDEs

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2008
Scott A. Sarra
Abstract Differentiation matrices associated with radial basis function (RBF) collocation methods often have eigenvalues with positive real parts of significant magnitude. This prevents the use of the methods for time-dependent problems, particulary if explicit time integration schemes are employed. In this work, accuracy and eigenvalue stability of symmetric and asymmetric RBF collocation methods are numerically explored for some model hyperbolic initial boundary value problems in one and two dimensions. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008 [source]