# Numerical Methods for Differential Equations in Physics

Introduction to    Python   Numpy    Matplotlib

I. Temporal Discretization   (slides)    (TD1)

II. Spatial Discretization   (slides)   (Intro to FFT)

III. Elliptic equations  (slides)

IV. Hyperbolic equations  (slides)

V. Curvilinear Coordinates  (slides)

VI. Computing Steady States   (slides)

VII. Computing Stability   (slides)

# Nonlinear Dynamics

I. Dynamical Systems (notes) (slides)

II. Convection and Lorenz Model (notes) (slides)

III. Symmetry (notes) (slides)

IV. Maps, Period Doubling and Floquet Theory (notes) (slides)

V. Quasiperiodicity and Intermittency (notes) (slides)

VI. Stripes, Patterns and Instabilities (notes) (slides)

VII. Reaction-Diffusion Equations (notes) (slides)

VIII. Hamiltonian Systems (notes) (slides)

IX. Codimension-Two Points (slides)

X. Practical Bifurcation Theory and Nonlinear Tourism (notes)

# Various explanations

Math or physics? Reflections on the nature of math, physics and natural philosophy. Normale Physics Review 7

Exercises sur les systèmes électoraux (pdf)
Exercises on voting theory (in French)

Linear stability analysis (pdf)
Global or not? That is the question.

Open Flows
Introduction to instablities in wall-bounded shear flows.

The Eckhaus instability (pdf)
Bifurcation analysis in a finite geometry.

Polar coordinates (pdf)
How to avoid singularities at the origin.

Rotating frames and fictitious forces (pdf)

Pressure gradient or flux (pdf)
Why it is necessary to impose one of these conditions in any periodic direction

Poisson operator in tensor product geometry (pdf)
How to apply or invert the Poisson operator with minimal time and storage

Non-dimensionsionalization of linear Faraday problem (pdf)

Notes on implicit time-stepping schemes (pdf)

Bifurcation Analysis for Time Steppers (pdf)
How to modify a time-stepping code to calculate steady states and traveling waves, and to perform linear stability analysis and compute optimal forcing

Recursion Relations and Influence Matrices (pdf)
Greens functions and the Sherman-Morrison-Woodbury formula.
How can systems of differential equations with coupled boundary conditions be solved?
When can matrices be reduced to banded form?
When do there exist recursion relations for differential operators?

Computational methods for MHD in a finite cylinder (pdf)

In preparation:

Jacobians for PDEs: matrix-free methods and preconditioning

Considering a control parameter, rather than the growth rate, as an eigenvalue.

### Ecole Polytechnique : Instabilités et Chaos (in French)Enseignement d’Approfondissement enMajeure de Mécanique

I. Systèmes Dynamiques (pdf)

II. Convection et Modèle de Lorenz (pdf)

III. Dédoublement de Période

IV. Ecoulements Ouverts (pdf)

V. Systèmes Hamiltoniens

### A Symmetry Primer for Fluid Mechanicians(under construction)

Reflection symmetry (pdf)

Rotations (pdf)

Complex eigenvalues (pdf)

Rectangular and centro-symmetry

Case study of channel flow