# Short Course on Geometric Numerical Integration of ODEs S15

*They throw geometry out the door, and it comes back through the window.*

-- H. G. Forder, Auckland 1973

In this course, we will cover basic techniques of structure-preserving or geometric numerical integration for ordinary differential equations. We will review the theoretical background of the continuous theory, that is basics of Lagrangian and Hamiltonian dynamics, variational principles, the concept of symplecticity, and how the Noether theorem connects symmetries and conservation laws. We will discuss basic symplectic integrators for Hamiltonian systems and explore several strategies for proving symplecticity. We will construct a discrete counterpart to Lagrangian mechanics, including Hamilton's action principle and the Noether theorem. At the end, we provide some practical tips on the efficient implementation of nonlinear implicit methods.

## Lecture Notes

There is a detailed scriptum as well as short lecture notes.

- Introduction
- Lagrangian and Hamiltonian Dynamics
- Lagrangian Dynamics
- Hamiltonian Dynamics
- Symplecticity
- Noether Theorem

- Symplectic Integrators
- Symplectic Euler
- Störmer-Verlet
- Symplectic-Partitioned Runge-Kutta Methods

- Variational Integrators
- Discrete Euler-Lagrange Equations
- Discrete Symplectic Form
- Discrete Noether Theorem

- Higher Order Integrators
- Composition Methods
- Variational Runge-Kutta Methods

- Implementation of Nonlinear Implicit Methods
- Fixpoint and Newton Iteration
- Starting Approximations
- Reducing Rounding Errors

## References

##### Main References

- Ernst Hairer, Christian Lubich and Gerhard Wanner. Geometric Numerical Integration. Springer, 2006.
- Jerrold E. Marsden and Matthew West. Discrete Mechanics and Variational Integrators. Acta Numerica Volume 10, page 357-514, 2001.

##### Differential Geometry

- John M. Lee. Introduction to Smooth Manifolds. Springer, 2013.
- Michael Spivak. A Comprehensive Introduction to Differential Geometry. Publish or Perish, 1999.
- Ralph Abraham, Jerrold E. Marsden, Tudor S. Ratiu. Manifolds, Tensor Analysis, and Applications. Springer, 1988.

##### Geometric Mechanics

- Jorge V. José and Eugene J. Saletan. Classical Dynamics. Cambridge University Press, 1998.
- Vladimir I. Arnol’d. Mathematical Methods of Classical Mechanics. Springer, 1989.
- Jerrold E. Marsden and Tudor S. Ratiu. Introduction to Mechanics and Symmetry. Springer, 1999.
- Ralph Abraham and Jerrold E. Marsden. Foundations of Mechanics. Addison-Wesley, 1987.

##### Numerical Analysis

- Ernst Hairer, Syvert P. Nørsett, and Gerhard Wanner. Solving Ordinary Differential Equations I: Nonstiff Problems. Springer, 1993.
- Ernst Hairer and Gerhard Wanner. Solving Ordinary Differential Equations II: Stiff and Differential- Algebraic Problems. Springer, 1996.
- Alfio Quarteroni, Riccardo Sacco, and Fausto Saleri. Numerical Mathematics. Springer, 2007.
- Kendall Atkinson and Weimin Han. Theoretical Numerical Analysis: A Functional Analysis Framework. Springer, 2009.