gpd-numerical-convergence

Run the computation across a geometric refinement sequence (e.g., doubling grid resolution over 3-5 levels), compute successive differences, and estimate the convergence order from Richardson ratios. The Skill automates this sweep, grading, and extrapolation for each identified parameter.

Richardson extrapolation combines results at two refinement levels to estimate the continuum limit when convergence follows a known power law O(h^p). Use it after confirming a consistent convergence order to obtain an extrapolated value with an error estimate.

Track conserved quantities like energy, momentum, or probability along the computed trajectory and quantify maximum and average drift. Bounded drift indicates a good symplectic integrator, while exponential drift signals numerical instability.

Oscillating errors typically indicate oscillatory integrands, catastrophic cancellation, or numerical instability rather than true convergence. The Skill detects these pitfalls explicitly and prescribes fixes such as Filon quadrature, higher precision arithmetic, or implicit integrators.

Yes. It compares explicit versus implicit integrators to detect stiffness and computes autocorrelation times to verify Monte Carlo sampling is decorrelated. It flags critical slowing down near phase transitions and recommends cluster algorithms or implicit methods.

Standard refinement comparisons fail for adaptive meshes, near critical phenomena with diverging correlation lengths, and when catastrophic cancellation amplifies relative error. These cases require finite-size scaling, tolerance-level studies, or reformulation to avoid subtracting large numbers.