Convergence Analysis for Selection Strategies of Set-Valued Runge-Kutta Methods

R. Baier: Convergence Analysis for Selection Strategies of Set-Valued Runge-Kutta Methods
Technical Report, 40 pages, University of Bayreuth, Bayreuth, Germany, June 2004

Keywords: set-valued Runge-Kutta methods; selection strategies; reachable sets; linear differential inclusions
Mathematics Subject Classification Code: 65L06 (54C65 65L05 93B03 34A60)
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Abstract:

A general framework for proving an order of convergence for set-valued Runge Kutta methods is given in the case of linear differential inclusions, if the attainable set at a given time should be approximated. The set-valued method is interpreted as a (set-valued) quadrature method with disturbed values for the fundamental solution at the nodes of the quadrature method. If the precision of the quadrature method and the order of the disturbances fit together, then an overall order of convergence could be guaranteed. The framework is applied to several Runge-Kutta methods up to order 4 with different selections strategies, i.e. piecewise constant, piecewise linear, two and more independent choices. Several numerical examples are calculated and the corresponding attainable sets are shown.

Contents:

1. Introduction
1.1 Differential Inclusions and Set-Valued Integral
1.2 Arithmetic Operations on Sets
1.3 Modulus of Smoothness
2. Quadrature and Combination Methods
2.1 Quadrature Methods
2.2 Quadrature Methods for the Approximation of Attainable Sets
2.3 Combination Methods
3. Set-Valued Runge-Kutta Methods
3.1 Euler's Method
3.2 Euler-Cauchy Method (or Heun's Method)
3.3 Modified Euler Method
3.4 Runge-Kutta (4)
4. Conclusions

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