Higher Order Modifying Integrators for Separable Equations

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Izmir 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 Gilles
  Vilmart, Numerical integrators based on modified
  differential equations, Math. Comp. 76 (2007)
  1941-1953.

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Outline
Izmir Institute of Technology

• Symplectic Euler Method and its Adjoint

• Lobatto IIIA-IIIB pair

• Midpoint Rule

• Application to Mechanical System

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• Consider the Initial Value Problem

• One Step Numerical Integrator

• Modified Differential Equation

• Modifiying Integrator (order r)

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• Need to find

recursively

• Midpoint rule as an example

• Apply this to Modified Differential
  Equation

Exact Solution
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Modified Vector Fields
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Main Equations

• Partitioned systems

• Separable systems

• Canonical Hamiltonian equations generated by
  Hamiltonian function

• Mechanical systems

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Symplectic Euler Method and its Adjoint
Partitioned systems

• Symplectic Euler Method

• Modified vector differential equation

• Modifiying Integrator

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Modified Vector Fields

• Order (h2)

• Order (h3)

• Hamiltonian Equation

• Modified Hamiltonian function

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Separable systems
Modified Vector Fields

• Order (h2)

• Order (h3)

• Modified Hamiltonian function

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Mechanical systems

• Modified vector differential equation with the
  Hamiltonian function

• Order (h2)

• Modifiying Integrator of Order (h2)

• Implicit in first, explicit in second argument

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Adjoint of Symplectic Euler Method

• Adjoint of Symplectic Euler Method

• Modified vector differential equation

• Order (h2)

• Order (h3)

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• 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)

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Mechanical systems

• Order (h2)

• Explicit in first and second arguments

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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.
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• Modified Hamiltonian function

Separable systems

• Modified vector fields

• Modified Hamiltonian function

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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

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Midpoint Rule

• ODE system

• Midpoint Rule

• Modified Vector Field

• Hamiltonian system

• Modified Hamilton Function

• Modified Differential Equation

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Separable systems

• Modified Differential Equation

Mechanical systems

• Modified Hamilton Function

• Useful Formulas

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• Application to Mechanical System

Double Well Potential

• Hamiltonian Function

• Modified Adjoint of Symplectic Euler Integrator
  of order 2

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Double Well Potential
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Double Well Potential
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Conclusion and Future Work
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Izmir Institute of Technology
References

• Chartier, Philippe Hairer, Ernst Vilmart,
  Gilles, 2007 Numerical 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
Demirs Master Thesis. ,
Thanks !