Construction of Runge–Kutta type methods for solving ordinary differential equations
In this paper, we study Runge–Kutta methods with continuous stage, introduced by Butcher in 1987. By setting the coefficients of this family of methods and choosing the appropriate numerical integration, we derive new classes of Runge–Kutta methods. Furthermore, we extend the W-transformation techni...
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| Published in: | Applied mathematics and computation Vol. 234; pp. 179 - 191 |
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| Abstract | In this paper, we study Runge–Kutta methods with continuous stage, introduced by Butcher in 1987. By setting the coefficients of this family of methods and choosing the appropriate numerical integration, we derive new classes of Runge–Kutta methods. Furthermore, we extend the W-transformation technique by permitting W to be a non-square matrix. This allows us to construct more high-order implicit Runge–Kutta methods with some geometric properties. Specially, we provide Runge–Kutta methods with continuous stage which are (conjugate) symplectic, symmetric or energy-preserving for solving Hamiltonian systems. We also present some numerical experiments to verify our results. |
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| AbstractList | In this paper, we study Runge–Kutta methods with continuous stage, introduced by Butcher in 1987. By setting the coefficients of this family of methods and choosing the appropriate numerical integration, we derive new classes of Runge–Kutta methods. Furthermore, we extend the W-transformation technique by permitting W to be a non-square matrix. This allows us to construct more high-order implicit Runge–Kutta methods with some geometric properties. Specially, we provide Runge–Kutta methods with continuous stage which are (conjugate) symplectic, symmetric or energy-preserving for solving Hamiltonian systems. We also present some numerical experiments to verify our results. |
| Author | Tang, Wensheng Sun, Yajuan |
| Author_xml | – sequence: 1 givenname: Wensheng surname: Tang fullname: Tang, Wensheng email: tangws@lsec.cc.ac.cn organization: College of Mathematics and Computational Science, Changsha University of Science and Technology, Changsha 410114, PR China – sequence: 2 givenname: Yajuan surname: Sun fullname: Sun, Yajuan email: sunyj@lsec.cc.ac.cn organization: LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, PR China |
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| Cites_doi | 10.1088/1751-8113/41/4/045206 10.1007/BF00945133 10.1088/0305-4470/29/13/006 10.1016/0375-9601(88)90773-6 10.1093/imanum/drs010 10.1063/1.4756391 10.1016/j.amc.2011.03.022 10.1137/0718074 10.1063/1.4756053 10.1007/BF02238798 10.1007/BF01954907 10.1016/j.amc.2012.08.062 10.1090/S0025-5718-1972-0315899-8 10.1063/1.4756051 10.1006/jcph.2000.6427 10.1016/j.amc.2012.01.074 10.1016/j.cam.2012.03.007 10.1090/S0025-5718-1964-0159424-9 10.1137/110856617 10.1051/m2an/2009020 10.1007/s00211-006-0003-8 10.1007/s10483-007-0809-y |
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| Keywords | Energy-preserving method Hamiltonian system (Conjugate) Symplectic method Runge–Kutta method with continuous stage Symmetric method W-transformation |
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| Snippet | In this paper, we study Runge–Kutta methods with continuous stage, introduced by Butcher in 1987. By setting the coefficients of this family of methods and... In this paper, we study Runge-Kutta methods with continuous stage, introduced by Butcher in 1987. By setting the coefficients of this family of methods and... |
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| SubjectTerms | (Conjugate) Symplectic method Computation Conjugates Construction Differential equations Energy-preserving method Hamiltonian system Mathematical models Numerical integration Runge-Kutta method Runge–Kutta method with continuous stage Symmetric method W-transformation |
| Title | Construction of Runge–Kutta type methods for solving ordinary differential equations |
| URI | https://dx.doi.org/10.1016/j.amc.2014.02.042 https://www.proquest.com/docview/1551044363 |
| Volume | 234 |
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