Lecturer: Georgios AKRIVIS, University of Ioannina, Ioannina, Greece Webpage:  http://www.cs.uoi.gr/~akrivis/ Title: NUMERICAL METHODS FOR INITIAL VALUE PROBLEMS Date and time: Mon, November 19 to Fri, November 23, 2012, 9:00 to 11:00 Abstract: We will consider the discretization of initial value problems for first order o.d.e’s,

We will first briefly review some basic stability properties of this problem, assuming that f satisfies either the Lipschitz condition or the one-sided Lipschitz condition. Then, the simplest scheme for (Ö), namely the Euler method, will be discussed. We will then focus on the two most popular classes of methods for initial value problems, namely the Runge–Kutta methods and multistep methods but will also briefly discuss Galerkin type schemes. Various stability as well as consistency properties of the schemes will be analyzed and advantages and drawbacks of each class will be discussed. Bibliography: [1] K. E. Atkinson, W. Han, D. E. Stewart: Numerical Solution of Ordinary Differential Equations. 3rd ed., Wiley, New York, 2009. [2] J. C. Butcher: Numerical Methods for Ordinary Differential Equations. 2nd ed., Wiley, 2008. [3] D. F. Griffiths, D. J. Higham: Numerical Methods for Ordinary Differential Equations. Initial Value Problems. Springer–Verlag, London, 2010. [4] E. Hairer, S. P. Nørsett, G. Wanner: Solving Ordinary Differential Equations I–Nonstiff Problems. 2nd revised ed., 1993, corr. 2nd printing, Springer–Verlag, Berlin, 2000. [5] E. Hairer, G. Wanner: Solving Ordinary Differential Equations II – Stiff and Differential–Algebraic Problems. 2nd revised ed., Springer–Verlag, Berlin, 2010. [6] P. Henrici: Discrete Variable Methods in Ordinary Differential Equations. Wiley, New York, 1962. [7] J. D. Lambert: Numerical Methods for Ordinary Differential Systems. Wiley, Chichester, 1991. [8] A. Tveito, H. P. Langtangen, B. F. Nielsen, X. Cai: Elements of Scientific Computing. Springer–Verlag, Berlin, 2011.

NUMERICAL METHODS

FOR INITIAL VALUE PROBLEMS

Abstract. We will consider the discretization of initial value problems for first ordero.d.e’s,

(!)

!

y! = f(t, y), a ! t ! b,

y(a) = y0.

We will first briefly review some basic stability properties of this problem, assumingthat f satisfies either the Lipschitz condition or the one-sided Lipschitz condition.Then, the simplest scheme for (!), namely the Euler method, will be discussed. We willthen focus on the two most popular classes of methods for initial value problems, namelythe Runge–Kutta methods and multistep methods but will also briefly discuss Galerkintype schemes. Various stability as well as consistency properties of the schemes will beanalyzed and advantages and drawbacks of each class will be discussed.

References

1. K. E. Atkinson, W. Han, D. E. Stewart: Numerical Solution of Ordinary Di!erential Equations.

3rd ed., Wiley, New York, 2009.

2. J. C. Butcher: Numerical Methods for Ordinary Di!erential Equations. 2nd ed., Wiley, 2008.

3. D. F. Gri!ths, D. J. Higham: Numerical Methods for Ordinary Di!erential Equations. Initial Value

Problems. Springer–Verlag, London, 2010.

4. E. Hairer, S. P. Nørsett, G. Wanner: Solving Ordinary Di!erential Equations I–Nonsti! Problems.

2nd revised ed., 1993, corr. 2nd printing, Springer–Verlag, Berlin, 2000.

5. E. Hairer, G. Wanner: Solving Ordinary Di!erential Equations II – Sti! and Di!erential–Algebraic

Problems. 2nd revised ed., Springer–Verlag, Berlin, 2010.

6. P. Henrici: Discrete Variable Methods in Ordinary Di!erential Equations. Wiley, New York, 1962.

7. J. D. Lambert: Numerical Methods for Ordinary Di!erential Systems. Wiley, Chichester, 1991.

8. A. Tveito, H. P. Langtangen, B. F. Nielsen, X. Cai: Elements of Scientific Computing. Springer–

Verlag, Berlin, 2011.

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