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The Continuous Extension Of The B-spline Linear Multistep Methodsfor Bvps On Non-uniform Meshes

机译：非均匀网格上Bv的B样条线性多步方法的连续扩展

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摘要

B-spline methods are Linear Multistep Methods based on B-splines which have good stability properties [F. Mazzia, A. Sestini. D. Trigiante, B-spline multistep methods and their continuous extensions, SIAM J. Numer Anal. 44 (5) (2006) 1954-1973] when used as Boundary Value Methods [L. Brugnano, D. Trigiante, Convergence and stability of boundary value methods for ordinary differential equations, J. Comput. Appl. Math. 66 (1-2) (1996) 97-109; L. Brugnano, D. Trigiante, Solving Differential Problems by Multistep Initial and Boundary Value Methods, Gordon and Breach Science Publishers, Amsterdam, 1998]. In addition, they have an important feature: if k is the number of steps, it is always possible to associate to the numerical solution a C~k spline of degree k + 1 collocating the differential equation at the mesh points. In this paper we introduce an efficient algorithm to compute this continuous extension in the general case of a non-uniform mesh and we prove that the spline shares the convergence order with the numerical solution. Some numerical results for boundary value problems are presented in order to show that the use of the information given by the continuous extension in the mesh selection strategy and in the Newton iteration makes more robust and efficient a Matlab code for the solution of BVPs.
机译：B样条方法是基于B样条的线性多步方法，具有良好的稳定性。 Mazzia，A。Sestini。 D. Trigiante，B样条多步方法及其连续扩展，SIAM J. Numer肛门。 [44（5）（2006）1954-1973]用作边界值方法[L. Brugnano，D。Trigiante，常微分方程的边值方法的收敛性和稳定性，J。Comput。应用数学。 66（1-2）（1996）97-109； L. Brugnano，D。Trigiante，《通过多步初始值和边值方法解决微分问题》，戈登和突破科学出版社，阿姆斯特丹，1998年。此外，它们还有一个重要特征：如果k是步数，则总是有可能将数值为k +1的C〜k样条与数值解相关联，从而将微分方程定位在网格点处。在本文中，我们介绍了一种在非均匀网格的一般情况下计算此连续扩展的有效算法，并证明了样条与数值解具有相同的收敛阶数。给出了一些关于边值问题的数值结果，以表明在网格选择策略和牛顿迭代中使用连续扩展给出的信息可以使BVP求解的Matlab代码更加健壮和高效。

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