To investigate a possible order reduction for general linear methods  we consider the Prothero-Robinson test problem  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 . It can be demonstrated that in the non-stiff case we have e[n] = O(h^p) as h → 0, if e = 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 = 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  E. Hairer and G. Wanner. Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer Verlag, Berlin, Heidelberg, New York, 1996.  Z. Jackiewicz. General Linear Methods for Ordinary Differential Equations. John Wiley, Hoboken, New Jersey, 2009.  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.

### Order reduction phenomenon for general linear methods

#### Abstract

To investigate a possible order reduction for general linear methods  we consider the Prothero-Robinson test problem  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 . It can be demonstrated that in the non-stiff case we have e[n] = O(h^p) as h → 0, if e = 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 = 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  E. Hairer and G. Wanner. Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer Verlag, Berlin, Heidelberg, New York, 1996.  Z. Jackiewicz. General Linear Methods for Ordinary Differential Equations. John Wiley, Hoboken, New Jersey, 2009.  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.
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Utilizza questo identificativo per citare o creare un link a questo documento: `http://hdl.handle.net/11386/4668060`
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