N th-order operator splitting schemes and nonreversible systems

This paper is concerned with partitioned Nth-order accurate split-operator schemes built using M distinct solution operators for semidiscrete equations of the form $\frac{d} {{dt}}u_j = f_j^{(1)} (\{ {u_k } \}) + \cdots + f_j^{(M)} (\{ {u_k } \})$, which arise, among others, from constant coefficient parabolic partial differential equations. We prove that, for every $N \geq 3$ and $M \geqslant 2$, each solution operator must be applied for at least one backward fractional time step during each complete time step. This result has important consequences for applications to the complex Ginzburg–Landau equation with periodic boundary conditions and other partial differential equations with both reversible and nonreversible components. Furthermore, for the special case of $N = 3$ and $M = 2$, we analytically determine all possible schemes.