Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations

@inproceedings{Hairer2004GeometricNI,
  title={Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations},
  author={Ernst Hairer and Christian Lubich and Gerhard Wanner},
  year={2004}
}
Important Aspects of Geometric Numerical Integration
TLDR
Using a modulated Fourier expansion, much insight can be gained for methods applied to problems where the high oscillations stem from a linear part of the vector field and where only one (or a few) high frequencies are present.
Numerical integration of variational equations.
  • Ch. SkokosE. Gerlach
  • Mathematics, Physics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2010
TLDR
It is found that the best numerical performance is exhibited by the "tangent map method," a scheme based on symplectic integration techniques which proves to be optimal in speed and accuracy.
Backward error analysis for conjugate symplectic methods
TLDR
This work shows how a blended version of variational and symplectic techniques can be used to compute modified symplectic and Hamiltonian structures systematically, provided that a variational formulation of the method is known.
A review of some geometric integrators
TLDR
The Discrete Exterior Calculus, which is a structure-preserving integrator based on differential geometry and exterior calculus, is presented, which leads to numerical schemes which preserve exactly the energy of the system.
Study of geometric integrators for differential equations
The aim of the work described in this thesis is the construction and the study of structure-preserving numerical integrators for differential equations, which share some geometric properties of the
Para-Hamiltonian form for General Autonomous ODE Systems: Introductory Results
We propose a new tool to deal with autonomous ODE systems for which the solution to the Hamiltonian inverse problem is not available in the usual, classical sense. Our approach allows a class of
Discrete Integrable Systems and Geometric Numerical Integration
This thesis deals with discrete integrable systems theory and modified Hamiltonian equations in the field of geometric numerical integration. Modified Hamiltonians are used to show that symplectic
...
...

References

SHOWING 1-10 OF 19 REFERENCES
600 XV.6.2 Linear Error Growth for Integrable Systems
    539 XIV.2.1 Canonical Transformation to Adiabatic Variables
      1.1 The Störmer-Verlet Method vs
      • XIII. Oscillatory Differential Equations with Constant High Frequencies . 471 XIII.1 Towards Longer Time Steps in Solving Oscillatory Equations of Motion
      Gautschi's and Deuflhard's Trigonometric Methods
        2.4 Energy Exchange between Stiff Components
          Underlying One-Step Method and Backward Error Analysis
            3.1 Modified Equation for Smooth Numerical Solutions
              Modified Hamiltonian of Multistep Methods
                1.2 Multistep Methods for Second Order Equations
                  478 XIII.2.1 Time Scales in the Fermi-Pasta-Ulam Problem
                    ...
                    ...