We study a new method in reducing the roundo error in computing derivatives using Chebyshev collocation methods. By using a grid mapping derived by Koslo and Tal-Ezer, and the proper choice of the parameter , the roundo error of the k-th derivative can be reduced from O(N2k) to O((N jln j)k), where is the machine precision and N is the number of collocation points. This drastic reduction of rou...

We study a new method in reducing the roundo error in computing derivatives using Chebyshev collocation methods. By using a grid mapping derived by Koslo and Tal-Ezer, and the proper choice of the parameter , the roundo error of the k-th derivative can be reduced from O(N) to O((N jln j)k), where is the machine precision and N is the number of collocation points. This drastic reduction of round...

In the field of scalar hyperbolic equations, we take into consideration the idea of building numerical schemes using two grids: the first one to represent the solution and the second one to collocate the equation. This approach is based on the experience already gained in the approximation of boundaryvalue elliptic-type problems (see for instance [1]), as well as in the field of functional ot i...

The well-known complex variable boundary element method (CVBEM) is extended for using collocation points not located at the usual boundary nodal point locations. In this work, several advancements to the implementation of the CVBEM are presented. The first advancement is enabling the CVBEM nodes to vary in location, impacting the modeling accuracy depending on chosen node locations. A second ad...

The method of orthogonal collocation for solving boundary value problems in ordinary diierential equation is described, and its relationship to classical nite diierence methods is illustrated. Numerical examples demonstrate the capabilities of software based on orthogonal collocation for analyzing nonlinear diierential equations. A new discretiza-tion method for the solution of elliptic partial...

The linear systems arising from MFS calculations share certain numerical effects with other systems involving radial basis functions. These effects concern approximation error and stability, which are closely related, and they can already be studied for simple interpolation problems without PDEs. In MFS calculations, they crucially depend on the position and density of the source points and the...

This paper deals with the numerical solution of boundary value problems of ordinary differential equations posed on infinite intervals. We cut the infinite interval at a finite, large enough point and insert additional, so-called asymptotic boundary conditions at the far (right) end and then solve the resulting two-point boundary value problem by an /1-stable symmetric collocation method. Probl...

This chapter describes an application of the recently proposed Modified Method of Fundamental Solutions (MMFS) to the potential flow problems. The solution in two dimensional Cartesian coordinates is represented in terms of the fundamental solution of the Laplace equation together with the first order polynomial augmentation. The collocation is used for determination of the expansion coefficien...

The electromagnetic field distribution as well as the resonating frequency of various modes in superconducting cavities are sensitive to small geometry deformations. The occurring variations are motivated by measurements of an available set of resonators from which we propose to extract a small number of relevant and independent deformations by using a truncated Karhunen–Loève expansion. The ra...

Two methods are presented for approximating the costate of optimal control problems in integral form using orthogonal collocation at Legendre–Gauss (LG) and Legendre–Gauss–Radau (LGR) points. It is shown that the derivative of the costate of the continuous-time optimal control problem is equal to the negative of the costate of the integral form of the continuous-time optimal control problem. Us...