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A note on continuous-stage Runge–Kutta methods

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  • Tang, Wensheng

Abstract

We provide a note on continuous-stage Runge–Kutta methods (csRK) for solving initial value problems of first-order ordinary differential equations. Such methods, as an interesting and creative extension of traditional Runge–Kutta (RK) methods, can give us a new perspective on RK discretization and it may enlarge the application of RK approximation theory in modern mathematics and engineering fields. A highlighted advantage of investigation of csRK methods is that we do not need to study the tedious solution of multi-variable nonlinear algebraic equations associated with order conditions. In this note, we will review, discuss and further promote the recently-developed csRK theory. In particular, we will place emphasis on geometric integrators including symplectic methods, symmetric methods and energy-preserving methods which play a central role in the field of geometric numerical integration.

Suggested Citation

  • Tang, Wensheng, 2018. "A note on continuous-stage Runge–Kutta methods," Applied Mathematics and Computation, Elsevier, vol. 339(C), pages 231-241.
    • Handle: RePEc:eee:apmaco:v:339:y:2018:i:c:p:231-241
    DOI: 10.1016/j.amc.2018.07.044
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    References listed on IDEAS

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    1. Tang, Wensheng & Lang, Guangming & Luo, Xuqiong, 2016. "Construction of symplectic (partitioned) Runge-Kutta methods with continuous stage," Applied Mathematics and Computation, Elsevier, vol. 286(C), pages 279-287.
    2. Tang, Wensheng & Zhang, Jingjing, 2018. "Symplecticity-preserving continuous-stage Runge–Kutta–Nyström methods," Applied Mathematics and Computation, Elsevier, vol. 323(C), pages 204-219.
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    Cited by:

    1. Tang, Wensheng & Zhang, Jingjing, 2019. "Symmetric integrators based on continuous-stage Runge–Kutta–Nyström methods for reversible systems," Applied Mathematics and Computation, Elsevier, vol. 361(C), pages 1-12.
    2. Tang, Wensheng & Sun, Yajuan & Zhang, Jingjing, 2019. "High order symplectic integrators based on continuous-stage Runge-Kutta-Nyström methods," Applied Mathematics and Computation, Elsevier, vol. 361(C), pages 670-679.

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