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Ordinary Differential Equations

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Numerical methods for

Ordinary Differential Equations

Prof. Marino Zennaro

1

1University of Trieste

Department of Mathematics and Geosciences Email: [email protected]

Timetable: 12 hrs. Course cancelled.

Course requirements: it is advisable to have attended a basic course in Numerical Analysis.

Examination and grading: Written exam.

SSD: MAT/08 Numerical Analysis

Aim: We present basic numerical methods for initial value problems in ordinary differential equations and we analyse their convergence properties.

Course contents:

Existence and uniqueness of the solution and continuous dependence on the data for the initial value problem y0(x) = f (x, y(x)), y(x0) = y0.

Classical Lipschitz constant and right hand side Lipschitz constant.

General one-step methods; explicit and implicit Runge-Kutta methods.

Definition of local truncation and discretization error for one-step methods and definition of consistency of order p. Convergence theorem with order p for one-step methods. Order condi- tions for Runge-Kutta methods. Order barriers for explicit and implicit methods.

Variable stepsize implementation. Embedded pairs of methods of Runge-Kutta-Fehlberg type.

References:

• E. Hairer, S.P. Norsett, G. Wanner: Solving Ordinary Differential Equations I, Nonstiff Problems, Springer-Verlag, Berlin, 1993

• J.C. Butcher: Numerical methods for ordinary differential equations. Second edition, John Wiley & Sons, Ltd., Chichester, 2008

• Lecture notes

MC-11

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