Symmetric-Adjoint and Symplectic-Adjoint Runge-Kutta Methods and Their Applications  

作　　者：Geng Sun Siqing Gan Hongyu Liu Zaijiu Shang 

机构地区：[1]Academy of Mathematics and Systems Science,Chinese Academy of Sciences,Beijing 100190,China [2]School of Mathematics and Statistics,HNP-LAMA,Central South University,Changsha,Hunan 410083,China [3]Department of Mathematics,City University of Hong Kong,Hong Kong,China

出　　处：《Numerical Mathematics(Theory,Methods and Applications)》2022年第2期304-335,共32页高等学校计算数学学报（英文版）

基　　金：supported by the NSF of China(No.11771436);The work of S.Gan was supported by the NSF of China,No.11971488;The work of H.Liu was supported by the Hong Kong RGC General Research Funds,12301218,12302919 and 12301420;The work of Z.Shang was supported by the NSF of China,No.11671392.

摘　　要：Symmetric and symplectic methods are classical notions in the theory of numerical methods for solving ordinary differential equations.They can generate numerical flows that respectively preserve the symmetry and symplecticity of the continuous flows in the phase space.This article is mainly concerned with the symmetric-adjoint and symplectic-adjoint Runge-Kutta methods as well as their applications.It is a continuation and an extension of the study in[14],where the authors introduced the notion of symplectic-adjoint method of a Runge-Kutta method and provided a simple way to construct symplectic partitioned Runge-Kutta methods via the symplectic-adjoint method.In this paper,we provide a more comprehensive and systematic study on the properties of the symmetric-adjoint and symplecticadjoint Runge-Kutta methods.These properties reveal some intrinsic connections among some classical Runge-Kutta methods.Moreover,those properties can be used to significantly simplify the order conditions and hence can be applied to the construction of high-order Runge-Kutta methods.As a specific and illustrating application,we construct a novel class of explicit Runge-Kutta methods of stage 6 and order 5.Finally,with the help of symplectic-adjoint method,we thereby obtain a new simple proof of the nonexistence of explicit Runge-Kutta method with stage 5 and order 5.

关 键 词：Runge-Kutta method SYMMETRIC SYMPLECTIC ADJOINT HIGH-ORDER explicit method 

分 类 号：O17[理学—数学]