Mathematics

Short notes on the methods behind Conatus: what is computed, in which form, and why. They are written for a reader who uses the library and wants to know what happens underneath; each one ends with the references it follows.

The notes describe the discrete problem Conatus actually solves. Where the native layer makes a choice for you (a tau placement, a gauge, the implicit/explicit split), the note says which rule it applies; explain() prints the same decisions for your own model.

• Spectral discretisation and the tau method
• Curvilinear geometries: regular bases
• Implicit–explicit time stepping
• Eigenproblems, residuals and the physical adjoint
• Invariant solutions: Newton–Krylov–hookstep
• Continuation and bifurcations
• Floquet–Hill analysis of time-periodic flows
• Non-modal analysis: transient growth
• Lyapunov exponents, covariant vectors and ensembles
• Symmetry subspaces, slices and phase-space geometry