基于径向基有限差分法求解带小粘性系数非齐次两点边值问题

A Finite Difference Method Based on Radial Basis Function for Non-Homogeneous Two-Point Boundary Value Problems with Small Viscosity Coefficient

摘要: 本文利用具有零次代数精度的一元径向基函数(Radial Basis Function，简记RBF)插值的Lagrange形式，给出在三等距节点的中心节点处逼近被插函数的一阶导数和二阶导数的有限差分(简记RBF-FD)公式。特别地，对二阶导函数径向基差分的逼近误差进行分析，给出了使其逼近阶达到最高的径向基函数的最佳参数值。然后，利用这些RBF-FD公式给出了求解带小粘性系数的非齐次两点边值问题的差分格式，使所构造格式的收敛阶是同节点模板的多项式差分格式收敛阶的2倍，而计算时间略有增加。数值实验表明所构造的RBF-FD格式在小粘性系数大于等于10⁻³的情况下能够保持四阶收敛速度。

Abstract: The paper uses the Lagrange’s form of one-dimensional radial basis function (abbr. RBF) interpo-lation with zero-degree algebraic precision to give the finite difference formulas (abbr. RBF-FD) for approximation of first derivative and second derivative of the interpolated function at the central node of the three equidistant nodes. In particular, the approximation error of second derivative of the RBF difference formula is analyzed, and the optimal parameter of the radial basis function, which makes the approximation order of RBF-FD reach the highest level, is obtained. Then, using these RBF-FD formulas, the RBF-FD scheme of the non-homogeneous two-point boundary value problem with small viscosity coefficient is given. The convergence order of the RBF-FD scheme constructed in this paper is two times of the polynomial finite difference scheme under the same node stencil. While, the calculating time of the RBF-FD scheme is only slightly increased. Numerical experiments indicate that the RBF-FD scheme can maintain the fourth-order convergence order under viscosity coefficient greater than or equal to 10⁻³.