TY - JOUR
T1 - Symmetric-Adjoint and Symplectic-Adjoint Runge-Kutta Methods and Their Applications
AU - Sun , Geng
AU - Gan , Siqing
AU - Liu , Hongyu
AU - Shang , Zaijiu
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 2
SP - 304
EP - 335
PY - 2022
DA - 2022/03
SN - 15
DO - http://doi.org/10.4208/nmtma.OA-2021-0097
UR - https://global-sci.org/intro/article_detail/nmtma/20354.html
KW - Runge-Kutta method, symmetric, symplectic, adjoint, high-order, explicit method.
AB - <p style="text-align: justify;">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 symplectic-adjoint 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.</p>