Symmetric general linear methods

The article considers symmetric general linear methods, a class of numerical time integration methods which, like symmetric Runge–Kutta methods, are applicable to general time-reversible differential equations, not just those derived from separable second-order problems. A definition of time-reversal symmetry is formulated for general linear methods, and criteria are found for the methods to be free of linear parasitism. It is shown that symmetric parasitism-free methods cannot be explicit, but such a method of order 4 is constructed with only one implicit stage. Several characterizations of symmetry are given, and connections are made with G-symplecticity. Symmetric methods are shown to be of even order, a suitable symmetric starting method is constructed and shown to be essentially unique. The underlying one-step method is shown to be time-symmetric.