On Multiscale RBF Collocation Methods for Solving the Monge–Ampère Equation

This paper considers some multiscale radial basis function collocation methods for solving the two-dimensional Monge–Ampère equation with Dirichlet boundary. We discuss and study the performance of the three kinds of multiscale methods. The first method is the cascadic meshfree method, which was pro...

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Bibliographic Details
Published in:Mathematical problems in engineering 2020, Vol.2020 (2020), p.1-10
Main Authors: Liu, Zhiyong, Xu, Qiuyan
Format: Article
Language:English
Subjects:
Accuracy
Algorithms
Collocation methods
Dirichlet problem
Engineering
Experimentation
Meshless methods
Methods
Monge-Ampere equation
Multiscale analysis
Nonlinear differential equations
Partial differential equations
Radial basis function
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Summary:This paper considers some multiscale radial basis function collocation methods for solving the two-dimensional Monge–Ampère equation with Dirichlet boundary. We discuss and study the performance of the three kinds of multiscale methods. The first method is the cascadic meshfree method, which was proposed by Liu and He (2013). The second method is the stationary multilevel method, which was proposed by Floater and Iske (1996), and is used to solve the fully nonlinear partial differential equation in the paper for the first time. The third is the hierarchical radial basis function method, which is constructed by employing successive refinement scattered data sets and scaled compactly supported radial basis functions with varying support radii. Compared with the first two methods, the hierarchical radial basis function method can not only solve the present problem on a single level with higher accuracy and lower computational cost but also produce highly sparse nonlinear discrete system. These observations are obtained by taking the direct approach of numerical experimentation.
ISSN:1024-123X
1563-5147
DOI:10.1155/2020/1748037