Structure-preserving algorithms for linear dynamical systems

Geng Sun

IM, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, P.R. China

sung@math.ac.cn

Poster

This paper is organized as follows. First it is shown that for source-free systems $\dot{y}=Ly$ if there exists a reversible matrix $P$ such that

$$P^{-1}LP=\left\{\begin{array}{lll} \hbox{diag}(0,\pm\lamb... ...\ &\hbox{as}&n=\hbox{even\ number}, \end{array}\right.$$

then symmetric and symplectic Runge-Kutta methods as well as symmetric partitioned RK methods with $\bar{b}=b_i,i=1,\cdots,s$ are volume-preserving. Second for a general linear dynamical system $\dot{y}=Ly$ first doing exponential transformation, and applying a modified $\theta$-method to the new system generated by the transformation can yield some first-order explicit structure-preserving schemes which can lead to $\hbox{det}(\frac{\partial y_1}{\partial y_0})=e^{h\hbox{tr}(L)}$, and then we compose the first-order schemes into arbitrarily high order explicit symmetric structure-preserving ones.