Finite difference schemes on quasi-uniform grids for BVPs on infinite intervals
Finite difference schemes on quasi-uniform grids for BVPs on infinite intervals
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DOI：
10.1016/j.cam.2014.02.036
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发表时间：
2012-11
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影响因子：
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通讯作者：
Riccardo Fazio;Alessandra Jannelli
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文献类型：
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作者：
Riccardo Fazio;Alessandra Jannelli
定义在无限区间上的边值问题的经典数值处理方法是用有限点处的适当边界条件代替无穷远处的边界条件，即所谓的截断边界。考虑到数值解的令人满意的精度的截断边界必须通过试算和误差来确定，这似乎是经典方法的最薄弱环节。另一方面，自由边界方法克服了对截断边界的先验定义的需要。事实上，在自由边界公式中，未知的自由边界可以用截断边界来识别，而未知的自由边界必须作为解的一部分。本文考虑了一种不同的方法来克服截断边界的引入，即定义在准均匀网格上的非标准有限差分格式。准均匀网格允许我们用有限个区间来描述无限区域。这样的网格的最后一个节点被放置在无穷大上，以便准确地考虑右边界条件。我们将所提出的方法应用于Falkner-Skan模型和一个在基础工程中感兴趣的问题。得到的数值结果与文献中的结果吻合得很好。此外，我们还提供了一种简单的方法来提高使用Richardson外推的数值结果的精度。最后，我们指出了一种可能的方法来将所提出的方法推广到定义在整个实线上的边值问题。
The classical numerical treatment of boundary value problems defined on infinite intervals is to replace the boundary conditions at infinity by suitable boundary conditions at a finite point, the so-called truncated boundary. A truncated boundary allowing for a satisfactory accuracy of the numerical solution has to be determined by trial and errors and this seems to be the weakest point of the classical approach. On the other hand, the free boundary approach overcomes the need for a priori definition of the truncated boundary. In fact, in a free boundary formulation the unknown free boundary can be identified with a truncated boundary and being unknown it has to be found as part of the solution.In this paper we consider a different way to overcome the introduction of a truncated boundary, namely non-standard finite difference schemes defined on quasi-uniform grids. A quasi-uniform grid allows us to describe the infinite domain by a finite number of intervals. The last node of such grid is placed on infinity so that right boundary conditions are taken into account exactly. We apply the proposed approach to the Falkner–Skan model and to a problem of interest in foundation engineering. The obtained numerical results are found in good agreement with those available in literature. Moreover, we provide a simple way to improve the accuracy of the numerical results using Richardson’s extrapolation. Finally, we indicate a possible way to extend the proposed approach to boundary value problems defined on the whole real line.