On the Order of General Linear Methods

General linear (GL) methods are numerical algorithms used to solve ODEs. The standard order conditions analysis involves the GL matrix itself and a starting procedure; however, a finishing method (F) is required to extract the actual ODE solution. The standard order analysis and stability are sufficient for the convergence of any GL method. Nonetheless, using a simple GL scheme, we show that the order definition may be too restrictive. Specifically, the order for GL methods with low order intermediate components may be underestimated. In this note we explore the order conditions for GL schemes and propose a new definition for characterizing the order of GL methods, which is focused on the final result – the outcome of F – and can provide more effective algebraic order conditions.