Structure-preserving algorithms for a class of dynamical systems

In this paper, we study structure-preserving algorithms for dynamical systems defined by ordinary differential equations in R-n. The equations are assumed to be of the form y = A(y) + D(y) + R(y), where A(y) is the conservative part subject to < A(y), y > = 0; D(y) is the damping part or the part describing the coexistence of damping and expanding; R(y) reflects strange phenomenon of the system. It is shown that the numerical solutions generated by the symplectic Runge-Kutta(SRK) methods with b(i) > 0 ( i = 1, ..., s) have long-time approximations to the exact ones, and these methods can describe the structural properties of the quadratic energy for these systems. Some numerical experiments and backward error analysis also show that these methods are better than other methods including the general algebraically stable Runge-Kutta(RK) methods.