Geometric Numerical Integration I S16

They throw geometry out the door, and it comes back through the window.
-- H. G. Forder, Auckland 1973

Applied mathematics is not engineering.
-- Paul Halmos, Mathematics Tomorrow 1981

Dates | Description | Prerequisites | Lecture Notes | Exercises | Python | References

Dates

18.-29.07., Tuesday until Friday 13:00-14:00, 14:15-15:15
Nishi-Waseda Campus, Building 51, Room 10-06

7月19日(火)3-4限
7月20日(水)3-4限
7月21日(木)3-4限
7月22日(金)3-4限

7月26日(火)3-4限
7月27日(水)3-4限
7月28日(木)3-4限
7月29日(金)3-4限

場所:51号館10-06教室

Description

In this course, we will cover basic techniques of structure-preserving or geometric numerical integration for ordinary and partial differential equations. In the first part (summer semester), we will review the theoretical background of the continuous theory, that is some geometry background, 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 strategies for proving symplecticity. We will see how volume preserving methods can be applied to more general than Hamiltonian systems. We will construct a discrete counterpart to Lagrangian mechanics, including Hamilton's action principle and the Noether theorem, leading to variational integrators. We will also cover integral preserving methods, which can be used to derive exactly energy preserving methods.

Prerequisites

Prerequisites are

Advantageous are

Familiarity with basic numerical methods (approximation, interpolation, numerical quadrature), basic differential geometry (manifolds, maps, flows, vector fields, differential forms, tangent and cotangent bundles) and classical mechanics (Lagrangian and Hamiltonian dynamics, action principles, symplecticity, Noether theorem) is of advantage but not required. All important concepts will be reviewed at the beginning of the course.

Lecture Notes

Script (modified 03.10.2016)

  1. Introduction (Slides, modified 19.07.2016)
  2. Some Numerical Analysis Background (Slides, modified 20.07.2016)
  3. Some Geometry Background (Slides, modified 21.07.2016)
  4. Lagrangian and Hamiltonian Dynamics
  5. Symplectic Integrators
  6. Volume Preserving Integrators
  7. Variational Integrators
  8. Integral Preserving Integrators
Additional Material
  1. Scientific Computing with Python
  2. Backward Error Analysis (modified 20.07.2016)

Exercises

  1. Symplectic Integrators (Solutions, Notebook)
  2. Volume Preserving Integrators (Solutions, Notebook)
  3. Variational Integrators (Solutions)
  4. Integral Preserving Integrators (Solutions)

Scientific Computing with Python

  1. Scientific Computing with Python
  2. Introduction to Python Programming
  3. NumPy
  4. SciPy
  5. Matplotlib
  6. SymPy
  7. HDF5
  8. Numba
  9. Cython

Download Notebooks

References

Additional references