A modified method of approximate particular solutions for solving linear and nonlinear PDEs
A modified method of approximate particular solutions for solving linear and nonlinear PDEs
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DOI：
10.1002/num.22161
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发表时间：
2017-11
影响因子：
3.9
通讯作者：
Guangming Yao;Chingshyang Chen;Hui Zheng
中科院分区：
文献类型：
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作者：
Guangming Yao;Chingshyang Chen;Hui Zheng
近似特定解法（MAPS）首先由 Chen 等人提出。载于 Chen、Fan 和 Wen，数值方法偏微分方程，28 (2012), 506–522。使用多重二次函数 (MQ) 和逆多重二次径向基函数 (RBF)。从那时起，许多常用 RBF 和微分算子的封闭形式特定解被导出。结果，MAPS 被扩展到 Matérn 和高斯 RBF。多调和样条（PS）由于其条件正定性和低精度而很少在 MAPS 中使用。 PS 的优点之一是无需考虑形状参数。在本文中，对 MAPS 进行了修改，以便可以更有效地使用 PS。在最初的 MAPS 中，使用了集成 RBF，即所谓的特定解决方案。使用 PS 时会添加额外的积分多项式基础。在修改后的 MAPS 中，附加多项式基直接添加到积分 RBF 中，无需积分。通过将 PS 的阶数增加到一定程度或增加配置点的数量，可以改善使用 PS 修改的 MAPS 的结果。在我们的大多数示例中，15 次或更小的多项式似乎效果良好。其他 RBF（例如 MQ）也可以在修改后的 MAPS 中使用。该方法的性能在许多示例上进行了测试，包括 2D 和 3D 中的线性和非线性问题。我们证明，一般来说，使用 PS 进行修改的 MAPS 在求解一般椭圆方程时比其他 RBF 更准确。© 2017 Wiley periodicals, Inc. NumerMethods Partial Differential Eq 33: 1839–1858, 2017
The method of approximate particular solutions (MAPS) was first proposed by Chen et al. in Chen, Fan, and Wen, Numer Methods Partial Differential Equations, 28 (2012), 506–522. using multiquadric (MQ) and inverse multiquadric radial basis functions (RBFs). Since then, the closed form particular solutions for many commonly used RBFs and differential operators have been derived. As a result, MAPS was extended to Matérn and Gaussian RBFs. Polyharmonic splines (PS) has rarely been used in MAPS due to its conditional positive definiteness and low accuracy. One advantage of PS is that there is no shape parameter to be taken care of. In this article, MAPS is modified so PS can be used more effectively. In the original MAPS, integrated RBFs, so called particular solutions, are used. An additional integrated polynomial basis is added when PS is used. In the modified MAPS, an additional polynomial basis is directly added to the integrated RBFs without integration. The results from the modified MAPS with PS can be improved by increasing the order of PS to a certain degree or by increasing the number of collocation points. A polynomial of degree 15 or less appeared to be working well in most of our examples. Other RBFs such as MQ can be utilized in the modified MAPS as well. The performance of the proposed method is tested on a number of examples including linear and nonlinear problems in 2D and 3D. We demonstrate that the modified MAPS with PS is, in general, more accurate than other RBFs for solving general elliptic equations.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1839–1858, 2017