解常微分方程的自适应分段多辛方法
Adaptive Piecewise Poly-Sinc Methods for Ordinary Differential Equations
摘要：
提出了一种新的基于Sinc点的常微分方程自适应分段逼近方法。自适应方法是一种分段配置法，它利用Poly-Sinc插值来达到近似的预定精度水平。我们的工作扩展了自适应分段Poly-Sinc方法的函数逼近，我们推导出一个先验的误差估计，我们的自适应方法，并显示其指数收敛的迭代次数。在这项工作中，我们显示了指数收敛的迭代次数的先验误差估计得到的分段配置法，提供了一个很好的估计的精确解的常微分方程在Sinc点存在。我们使用统计方法进行分区细化。在正则和刚性常微分方程上验证了自适应贪婪分段Poly-Sinc算法。
英文摘要：
We propose a new method of adaptive piecewise approximation based on Sinc points for ordinary differential equations. The adaptive method is a piecewise collocation method which utilizes Poly-Sinc interpolation to reach a preset level of accuracy for the approximation. Our work extends the adaptive piecewise Poly-Sinc method to function approximation, for which we derived an a priori error estimate for our adaptive method and showed its exponential convergence in the number of iterations. In this work, we show the exponential convergence in the number of iterations of the a priori error estimate obtained from the piecewise collocation method, provided that a good estimate of the exact solution of the ordinary differential equation at the Sinc points exists. We use a statistical approach for partition refinement. The adaptive greedy piecewise Poly-Sinc algorithm is validated on regular and stiff ordinary differential equations.
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