Volume 12, Issue 5
The Modified Localized Method of Approximated Particular Solutions for Linear and Nonlinear Convection-Diffusion-Reaction PDEs

Wen Li, Kalani Rubasinghe, Guangming Yao & L. H. Kuo

Adv. Appl. Math. Mech., 12 (2020), pp. 1113-1136.

Published online: 2020-07

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  • Abstract

In this paper, a kernel based method, the modified localized method of approximated particular solutions (MLMAPS) [16,23] is utilized to solve unsteady-state linear and nonlinear diffusion-reaction PDEs with or without convections. The time-space and spatial space are discretized by the higher-order Houbolt method with various time step sizes and the MLMAPS, respectively. The local truncation error associated with the time discretization is $\mathcal{O}(h^4)$, where $h$ is the largest time step size used. The spatial domain is then treated by a special kernel, the integrated polyharmonic splines kernels together with low-order polynomial basis. Typical computational algorithms require a trade off between accuracy and rate of convergency. However, the experimental analysis has shown high accuracy and fast convergence of the proposed method.

  • Keywords

Polyharmonic spline, Houbolt method, time-dependent PDEs, method of approximated particular solutions, MLMAPS, convection-diffusion-reaction, nonlinear, kernel methods.

  • AMS Subject Headings

65N35, 65N99

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-12-1113, author = {Wen Li , and Kalani Rubasinghe , and Guangming Yao , and L. H. Kuo , }, title = {The Modified Localized Method of Approximated Particular Solutions for Linear and Nonlinear Convection-Diffusion-Reaction PDEs}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2020}, volume = {12}, number = {5}, pages = {1113--1136}, abstract = {

In this paper, a kernel based method, the modified localized method of approximated particular solutions (MLMAPS) [16,23] is utilized to solve unsteady-state linear and nonlinear diffusion-reaction PDEs with or without convections. The time-space and spatial space are discretized by the higher-order Houbolt method with various time step sizes and the MLMAPS, respectively. The local truncation error associated with the time discretization is $\mathcal{O}(h^4)$, where $h$ is the largest time step size used. The spatial domain is then treated by a special kernel, the integrated polyharmonic splines kernels together with low-order polynomial basis. Typical computational algorithms require a trade off between accuracy and rate of convergency. However, the experimental analysis has shown high accuracy and fast convergence of the proposed method.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2019-0033}, url = {http://global-sci.org/intro/article_detail/aamm/17742.html} }
TY - JOUR T1 - The Modified Localized Method of Approximated Particular Solutions for Linear and Nonlinear Convection-Diffusion-Reaction PDEs AU - Wen Li , AU - Kalani Rubasinghe , AU - Guangming Yao , AU - L. H. Kuo , JO - Advances in Applied Mathematics and Mechanics VL - 5 SP - 1113 EP - 1136 PY - 2020 DA - 2020/07 SN - 12 DO - http://doi.org/10.4208/aamm.OA-2019-0033 UR - https://global-sci.org/intro/article_detail/aamm/17742.html KW - Polyharmonic spline, Houbolt method, time-dependent PDEs, method of approximated particular solutions, MLMAPS, convection-diffusion-reaction, nonlinear, kernel methods. AB -

In this paper, a kernel based method, the modified localized method of approximated particular solutions (MLMAPS) [16,23] is utilized to solve unsteady-state linear and nonlinear diffusion-reaction PDEs with or without convections. The time-space and spatial space are discretized by the higher-order Houbolt method with various time step sizes and the MLMAPS, respectively. The local truncation error associated with the time discretization is $\mathcal{O}(h^4)$, where $h$ is the largest time step size used. The spatial domain is then treated by a special kernel, the integrated polyharmonic splines kernels together with low-order polynomial basis. Typical computational algorithms require a trade off between accuracy and rate of convergency. However, the experimental analysis has shown high accuracy and fast convergence of the proposed method.

Wen Li, Kalani Rubasinghe, Guangming Yao & L. H. Kuo. (2020). The Modified Localized Method of Approximated Particular Solutions for Linear and Nonlinear Convection-Diffusion-Reaction PDEs. Advances in Applied Mathematics and Mechanics. 12 (5). 1113-1136. doi:10.4208/aamm.OA-2019-0033
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