A lot of methods for solving boundary value problems using arbitrary grids, such as SDI (scattered data interpolation) and SPH (smoothed particle hydrodynamics), use families of atomic radial basis functions that depend on parameters to improve the accuracy of calculations. Functions of this kind are commonly called "shape functions". When polynomials or polynomial splines are used as such functions, they are called "basis functions". The term "radial" means that the carrier of the function is a disk or layer. The term "atomic" means that the support of the function is limited, ie the function is finite. In most cases, the term "finite" is used in English-language publications. The article presents an algorithm for constructing such a function, which is the solution of the functional-differential equation
where - circle of radius r: , and . The function generated by this equation has two parameters: r and . Variation of these parameters allows to reduce the error in the calculations of the Poisson boundary value problem by several times. The theorem on the existence of such an unambiguous function is proved in the article. The proof of the theorem allows us to construct one-dimensional Fourier transform of this function in the form , where . Previously, function was calculated using its Taylor approximation (at ), and at – using the asymptotic Hankel approximation of the function . Thus in a circle of a point a fairly large error was found. Therefore, the calculation of the function in the range was carried out by Chebyshev approximation of this function in the range . Chebyshev coefficients (calculated in the Maple 18 system with an accuracy of 26 decimal digits) and the range were chosen by an experiment aimed at minimizing the overall error in calculating of the function . Thanks to the use of the Chebyshev approximation, the obtained function has more than twice less error than calculated by the previous algorithm. Arbitrary value of the function is calculated using a six-point Aitken scheme, which can be considered (to some extent) a smoothing filter. The use of Aitken's six-point scheme introduces an error equal to 6% of the total function calculation error , but helps to save a lot of time in the formation of ARBF in solving boundary value problems using the method of collocation.
A Cartesian-grid collocation technique with integrated radial basis functions for mixed boundary value problems
