Efficient Fourth-Order Weights in Kernel-Type Methods without Increasing the Stencil Size with an Application in a Time-Dependent Fractional PDE Problem
An effective strategy to enhance the convergence order of nodal approximations in interpolation or PDE problems is to increase the size of the stencil, albeit at the cost of increased computational burden. In this study, our goal is to improve the convergence orders for approximating the first and s...
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| Format: | Article |
| Language: | English |
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MDPI AG
2024-04-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/12/7/1121 |
| _version_ | 1797212244444971008 |
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| author | Tao Liu Stanford Shateyi |
| author_facet | Tao Liu Stanford Shateyi |
| author_sort | Tao Liu |
| collection | DOAJ |
| description | An effective strategy to enhance the convergence order of nodal approximations in interpolation or PDE problems is to increase the size of the stencil, albeit at the cost of increased computational burden. In this study, our goal is to improve the convergence orders for approximating the first and second derivatives of sufficiently differentiable functions using the radial basis function-generated Hermite finite-difference (RBF-HFD) scheme. By utilizing only three equally spaced points in 1D, we are able to boost the convergence rate to four. Extensive tests have been conducted to demonstrate the effectiveness of the proposed theoretical weighting coefficients in solving interpolation and PDE problems. |
| first_indexed | 2024-04-24T10:39:18Z |
| format | Article |
| id | doaj.art-998717d9d50541c89ec48c410a901393 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-04-24T10:39:18Z |
| publishDate | 2024-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-998717d9d50541c89ec48c410a9013932024-04-12T13:22:57ZengMDPI AGMathematics2227-73902024-04-01127112110.3390/math12071121Efficient Fourth-Order Weights in Kernel-Type Methods without Increasing the Stencil Size with an Application in a Time-Dependent Fractional PDE ProblemTao Liu0Stanford Shateyi1School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, ChinaDepartment of Mathematics and Applied Mathematics, School of Mathematical and Natural Sciences, University of Venda, P. Bag X5050, Thohoyandou 0950, South AfricaAn effective strategy to enhance the convergence order of nodal approximations in interpolation or PDE problems is to increase the size of the stencil, albeit at the cost of increased computational burden. In this study, our goal is to improve the convergence orders for approximating the first and second derivatives of sufficiently differentiable functions using the radial basis function-generated Hermite finite-difference (RBF-HFD) scheme. By utilizing only three equally spaced points in 1D, we are able to boost the convergence rate to four. Extensive tests have been conducted to demonstrate the effectiveness of the proposed theoretical weighting coefficients in solving interpolation and PDE problems.https://www.mdpi.com/2227-7390/12/7/1121radial basis function (RBF)convergence orderHermite finite difference (HFD)analytical weightsfractional PDE |
| spellingShingle | Tao Liu Stanford Shateyi Efficient Fourth-Order Weights in Kernel-Type Methods without Increasing the Stencil Size with an Application in a Time-Dependent Fractional PDE Problem Mathematics radial basis function (RBF) convergence order Hermite finite difference (HFD) analytical weights fractional PDE |
| title | Efficient Fourth-Order Weights in Kernel-Type Methods without Increasing the Stencil Size with an Application in a Time-Dependent Fractional PDE Problem |
| title_full | Efficient Fourth-Order Weights in Kernel-Type Methods without Increasing the Stencil Size with an Application in a Time-Dependent Fractional PDE Problem |
| title_fullStr | Efficient Fourth-Order Weights in Kernel-Type Methods without Increasing the Stencil Size with an Application in a Time-Dependent Fractional PDE Problem |
| title_full_unstemmed | Efficient Fourth-Order Weights in Kernel-Type Methods without Increasing the Stencil Size with an Application in a Time-Dependent Fractional PDE Problem |
| title_short | Efficient Fourth-Order Weights in Kernel-Type Methods without Increasing the Stencil Size with an Application in a Time-Dependent Fractional PDE Problem |
| title_sort | efficient fourth order weights in kernel type methods without increasing the stencil size with an application in a time dependent fractional pde problem |
| topic | radial basis function (RBF) convergence order Hermite finite difference (HFD) analytical weights fractional PDE |
| url | https://www.mdpi.com/2227-7390/12/7/1121 |
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