numerical-integration

Time integration method selection depends on stiffness: explicit methods like Runge-Kutta suit non-stiff problems with fast dynamics, while implicit multistep schemes handle stiff systems where fast and slow timescales coexist. Use the integrator selector with the --stiff flag to match your problem's characteristics and accuracy requirements.

Adaptive time stepping automatically adjusts step size based on estimated local error, keeping accuracy within a tolerance while reducing unnecessary computation. It's essential for PDEs with varying dynamics—reaction-diffusion systems, for example—where fixed steps either waste time or accumulate error.

IMEX (implicit-explicit) schemes treat stiff terms implicitly and non-stiff terms explicitly, avoiding the cost of fully implicit solves. Plan your splitting by identifying which terms dominate stiffness, then use the IMEX planning feature to configure the split and select an appropriate IMEX method like SBDF2.

Error control uses error norms and step controllers to estimate local truncation error and adjust step size accordingly. Configure tolerance via error-norm thresholds and integrator order; tighter tolerances improve accuracy but increase cost, while looser tolerances reduce computation at the expense of precision.

Yes, this Skill requires only Python 3.8+ and NumPy with minimal dependencies, making it lightweight and portable. CLI scripts emit structured JSON outputs compatible with downstream workflows, so you can integrate step recommendations and error metrics into larger simulation pipelines.

Stiff problems contain both fast and slow timescales; explicit methods become inefficient because they must use tiny steps to resolve fast dynamics. Non-stiff problems have well-separated timescales or smooth behavior; explicit Runge-Kutta methods are faster. Use the stiff flag to select appropriate schemes.