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Higher Order Modifying Integrators for Separable Equations Presented by

İzmir Institute of Technology. Higher Order Modifying Integrators for Separable Equations Presented by Assoc. Prof. Dr. Gamze Tanoglu Joint work with Roman Kozlov IYTE Department of Mathematics. Splitting Methods in Time Integration, Innsbruck, 08. h. Objective

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Higher Order Modifying Integrators for Separable Equations Presented by

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  1. İzmir Institute of Technology Higher Order Modifying Integrators for Separable Equations Presented by Assoc. Prof. Dr.Gamze Tanoglu Joint work with Roman Kozlov IYTE Department of Mathematics Splitting Methods in Time Integration, Innsbruck, 08 h

  2. Objective • Develop a higher order Numerical Integrators which preserve structural properties of the differential equations based on modified vector field. • Background • Philippe Chartier, Ernst Hairer, and GillesVilmart, Numerical integratorsbased on modified differential equations, Math. Comp. 76 (2007) 1941-1953.

  3. İzmir Institute of Technology Outline • Symplectic Euler Method and its Adjoint • Lobatto IIIA-IIIB pair • Midpoint Rule • Application to Mechanical System

  4. Consider the Initial Value Problem • One Step Numerical Integrator • Modified Differential Equation • Modifiying Integrator (order r)

  5. Need to find recursively • Midpoint rule as an example : • Apply this to Modified Differential Equation Exact Solution

  6. Modified Vector Fields :

  7. Main Equations • Partitionedsystems • Separable systems • Canonical Hamiltonian equations generated by Hamiltonian function • Mechanical systems

  8. Symplectic Euler Method and its Adjoint Partitioned systems • Symplectic Euler Method • Modified vector differential equation • Modifiying Integrator

  9. Modified Vector Fields : • Order (h2) • Order (h3) • Hamiltonian Equation • Modified Hamiltonian function

  10. Separable systems Modified Vector Fields : • Order (h2) • Order (h3) • Modified Hamiltonian function

  11. Mechanical systems • Modified vector differential equation with the Hamiltonian function • Order (h2) • Modifiying Integratorof Order (h2) • Implicit in first, explicit in second argument

  12. Adjoint of Symplectic Euler Method • Adjoint of Symplectic Euler Method • Modified vector differential equation • Order (h2) • Order (h3)

  13. Modified Hamiltonian function Remark: • Splitting Methods Composition of SE and SE∗ ⇒ symplectic. Results: Composition of SE1 and SE1∗ ⇒ Order 2 Composition of SE2 and SE2∗ ⇒ Order 2 Composition of SE3 and SE3∗ ⇒ Order 4 (Separable system)

  14. Mechanical systems • Order (h2) • Explicit in first and second arguments

  15. Lobatto IIIA-IIIB pair • Modified vector differential equation Lemma : Application of the Lobatto IIIA-IIIB pair of the second order to the modified differential equation gives a numerical method of order 4.

  16. Modified Hamiltonian function Separable systems • Modified vector fields • Modified Hamiltonian function

  17. Mechanical systems • Modified Hamiltonian function • Modified vector differential equation • Application to Lobatto IIIA-IIIB pair of order 2 • First and second stages are implicit and third is explicit

  18. Midpoint Rule • ODE system • MidpointRule • Modified Vector Field • Hamiltonian system • Modified Hamilton Function • Modified Differential Equation

  19. Separable systems • Modified Differential Equation Mechanical systems • ModifiedHamilton Function • Useful Formulas

  20. Application to Mechanical System Double Well Potential • Hamiltonian Function • Modified Adjoint of Symplectic Euler Integrator of order 2

  21. Double Well Potential

  22. Double Well Potential

  23. Conclusion and Future Work

  24. İzmir Institute of Technology References • Chartier, Philippe; Hairer, Ernst; Vilmart, Gilles, 2007Numerical integrators based on modified differential equations,Math. Comp., 76, no. 260 1941--1953 . • Philippe Chartier, Ernst Hairer and Gilles Vilmart , 2007 Modified differential equations, ESAIM: Proceedings , Vol 21, 16-20. • Hairer, Ernst, Lubich, Christian and Wanner, Gerhard 2006, Geometric numerical integration, Structure--preserving algorithms for ordinary differential equations, (Berlin: Springer--Verlag) Acknowlegments The results present in this talk is obtained during the visit of the Roman Kozlov to Izmir Inst. Of Tech. in July, 2008. His visit was supported by the Scientific and Techonological Researh Council of Turkey (TUBITAK). Some part of this work will be submitted to IYTE Graduate School as a Duygu Demir`s Master Thesis. , Thanks !

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