Numerical Continuation of Patterns

Numerical continuation of MDP (David Lloyd) 

Day 1 topics: 

1. Numerical continuation of steady states of ODES - definition of steady state, newton’s method, pseudo-arclength continuation. Stability calculation. Will do this for Swift-Hohenberg equation and Reaction-Diffusion systems

2. Numerical continuation of periodic states of ODES - Differentiation matrices (finite differences and Fourier pseudo-spectral methods), why Newton’s method fails (continuous symmetries and a non-trivial kernel of the linearisation), even restriction or phase condition method of regularisation, embedding in the continuation framework. Co-periodic stability analysis. 

3. Numerical continuation of localised states of ODES - Continuous symmetries, phase condition regularisation or even restriction approach. Differentiation matrices and numerical continuation

Lab 1: Implementation of Day 1 topics 1-3. 

Day 2 topics: 

1. Floquet/Bloch stability of stripes extended in 2D. Application Swift-Hohenberg equation. Cover Eckhaus and Zig-zag instability

2. Computing 2D localised structures - Differentiation matrices, kronecker products as a simple way to implement this. Localised patterns on the periodic strip and fully localised 2D patterns. Numerical continuation.

3. Far-field core decomposition method in 1D for computing stationary PtoE or PtoP fronts. Application Swift-Hohenberg equation 

Lab 2: Implementation of Day 2 topics 1-3.