Accelerated Runge-Kutta Methods
Standard Runge-Kutta methods are explicit, one-step, and generally constant step-size numerical integrators for the solution of initial value problems. Such integration schemes of orders 3, 4, and 5 require 3, 4, and 6 function evaluations per time step of integration, respectively. In this paper, w...
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| Format: | Article |
| Language: | English |
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Wiley
2008-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2008/790619 |
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| _version_ | 1850221320162246656 |
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| author | Firdaus E. Udwadia Artin Farahani |
| author_facet | Firdaus E. Udwadia Artin Farahani |
| author_sort | Firdaus E. Udwadia |
| collection | DOAJ |
| description | Standard Runge-Kutta methods are explicit, one-step, and generally constant step-size numerical integrators for the solution of initial value problems. Such integration schemes of orders 3, 4, and 5 require 3, 4, and 6 function evaluations per time step of integration, respectively. In this paper, we propose a set of simple, explicit, and constant step-size Accerelated-Runge-Kutta methods that are two-step in nature. For orders 3, 4, and 5, they require only 2, 3, and 5 function evaluations per time step, respectively. Therefore, they are more computationally efficient at achieving the same order of local accuracy. We present here the derivation and optimization of these accelerated integration methods. We include the proof of convergence and stability under certain conditions as well as stability regions for finite step sizes. Several numerical examples are provided to illustrate the accuracy, stability, and efficiency of the proposed methods in comparison with standard Runge-Kutta methods. |
| format | Article |
| id | doaj-art-6cb8d4e2722a4b20b89ea18977fbca6e |
| institution | OA Journals |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2008-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-6cb8d4e2722a4b20b89ea18977fbca6e2025-08-20T02:06:44ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2008-01-01200810.1155/2008/790619790619Accelerated Runge-Kutta MethodsFirdaus E. Udwadia0Artin Farahani1Departments of Aerospace and Mechanical Engineering, Civil Engineering, Mathematics, and Information and Operations Management, 430K Olin Hall, University of Southern California, Los Angeles, CA 90089-1453, USADepartment of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA 90089-1453, USAStandard Runge-Kutta methods are explicit, one-step, and generally constant step-size numerical integrators for the solution of initial value problems. Such integration schemes of orders 3, 4, and 5 require 3, 4, and 6 function evaluations per time step of integration, respectively. In this paper, we propose a set of simple, explicit, and constant step-size Accerelated-Runge-Kutta methods that are two-step in nature. For orders 3, 4, and 5, they require only 2, 3, and 5 function evaluations per time step, respectively. Therefore, they are more computationally efficient at achieving the same order of local accuracy. We present here the derivation and optimization of these accelerated integration methods. We include the proof of convergence and stability under certain conditions as well as stability regions for finite step sizes. Several numerical examples are provided to illustrate the accuracy, stability, and efficiency of the proposed methods in comparison with standard Runge-Kutta methods.http://dx.doi.org/10.1155/2008/790619 |
| spellingShingle | Firdaus E. Udwadia Artin Farahani Accelerated Runge-Kutta Methods Discrete Dynamics in Nature and Society |
| title | Accelerated Runge-Kutta Methods |
| title_full | Accelerated Runge-Kutta Methods |
| title_fullStr | Accelerated Runge-Kutta Methods |
| title_full_unstemmed | Accelerated Runge-Kutta Methods |
| title_short | Accelerated Runge-Kutta Methods |
| title_sort | accelerated runge kutta methods |
| url | http://dx.doi.org/10.1155/2008/790619 |
| work_keys_str_mv | AT firdauseudwadia acceleratedrungekuttamethods AT artinfarahani acceleratedrungekuttamethods |