ON THE PRESERVATION OF SECOND INTEGRALS BY RUNGE-KUTTA METHODS

被引:1
作者
Tapley, Benjamin K. [1 ,2 ]
机构
[1] SINTEF Digital, Dept Math & Cybernet, N-0373 Oslo, Norway
[2] Norwegian Univ Sci & Technol, Dept Math Sci, N-7491 Trondheim, Norway
来源
JOURNAL OF COMPUTATIONAL DYNAMICS | 2023年 / 10卷 / 02期
基金
欧盟地平线“2020”;
关键词
Key veords and phrases; Dynamical systems; Runge-Kutta methods; second integrals; Darboux polynomials; discrete dynamical systems; SYSTEMS;
D O I
10.3934/jcd.2023001
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
One can elucidate integrability properties of ordinary differential equations (ODEs) by knowing the existence of second integrals (also known as weak integrals or Darboux polynomials for polynomial ODEs). However, little is known about how they are preserved, if at all, under numerical methods. Here, we leverage the recently discovered theory of discrete second integrals to show novel results about Runge-Kutta methods. In particular, we show that any Runge-Kutta method preserves all affine second integrals but cannot preserve all quadratic second integrals of an ODE. A number of interesting corollaries are also discussed, such as the preservation of certain rational in-tegrals by Runge-Kutta methods. The special case of affine second integrals with constant cofactor are also discussed as well the preservation of third and higher integrals.
引用
收藏
页码:304 / 322
页数:19
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