The existence of symplectic general linear methods
The paper obtains a criterion that any general linear integration method must satisfy if it is symplectic. After deriving a criterion for symplectic behavior based on a standard conservative problem of the form, \[ \begin{cases} z_1'(t)= -\alpha(z_3,z_4,\dots,z_N)z_2(t), \\ z_2'(t)= \alpha(z_3,z_4,\dots,z_N)z_1(t), \\ z_i'(t)=f_i(z_3,z_4,\dots,z_N)z_1(t), \quad i=3,4\dots,N, \end{cases} \] where the linear factor \(\alpha\) can take on different values at the stages of the numerical method, the authors give a linear algebraic condition satisfied by all symplectic general methods. As a result of this discussion, the authors address the reducibility problem of a general linear method with \(s\) stages and \(r\) inputs. Although the question of full reducibility is not considered in general, the main contribution of the paper is to prove that if a method with \(r>1\) exhibits symplectic behavior, then it is reducible to a method with \(r=1\). In other words, the main theorem states that a stable, irreducible, general linear method with \(r>1\) cannot be symplectic, and so linking the criterion of the paper to that for Runge-Kutta methods.
scientific article; zbMATH DE number 5571631