anthropocene◆
Computational Physics / ODE Solvers, One by One
◆ advanced30 mindraft

Stiffness, and evaluating the slope where you are going

When the fastest mode in the system forces a step size thousands of times smaller than accuracy needs. The fix costs a nonlinear solve per step and is worth it.

assumes you already have

Planned outline

  1. A concrete stiff system: two modes with timescales separated by 10^6.
  2. Backward Euler, and why the unknown appearing on both sides is the whole point.
  3. Solving the implicit step with Newton, and why fixed-point iteration fails exactly when you need it.
  4. Stability regions in the complex plane; A-stability and L-stability.
  5. The implicit trapezoid rings on stiff modes while backward Euler damps them - L-stability made visible.
  6. BDF methods and what production stiff solvers actually use.

Interactives planned

Each numbered point above gets at least one thing you can manipulate, not a figure. See AGENTS.md for the widget vocabulary and the rule that every lesson ships at least one graded interaction and one recall card.