Order reduction phenomenon for general linear methods

To investigate a possible order reduction for general linear methods [2] we consider the Prothero-Robinson test problem [3] y'(t) = λ y(t) − ϕ(t)+ ϕ'(t), t ∈ [t0, T], y(t0) = ϕ(t0), Re(λ) ≤ 0, with exact solution y(t) = ϕ(t). Let h be a stepsize and z = hλ. We are interested in the behavior of the global error e[n] at n-th step as h → 0 and z = O(h), which corresponds to the classical non-stiff case, and as h → 0 and z → −∞, which corresponds to the stiff case, i.e., when Re(λ) ≪ 0, compare [1]. It can be demonstrated that in the non-stiff case we have e[n] = O(h^p) as h → 0, if e[0] = O(h^p) and the general linear method has order p, regardless of the stage order q. In the stiff case we assume that the general linear method has order p and stage order q ≤ p and e[0] = O(h^p) as h → 0. Then e[n] = O(h^q) + O(h^p) as h → 0 and z → −∞. The global error estimate can be improved for general linear methods with so-called Runge–Kutta stability which are A-stable: e[n] = O(h^(q+1)) + O(h^p) as h → 0 and z → −∞. Theoretical results are illustrated by examples of methods and numerical experiments. REFERENCES [1] E. Hairer and G. Wanner. Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer Verlag, Berlin, Heidelberg, New York, 1996. [2] Z. Jackiewicz. General Linear Methods for Ordinary Differential Equations. John Wiley, Hoboken, New Jersey, 2009. [3] A. Prothero and A. Robinson. On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations. Math. Comput., 28: 145–162, 1974