numerical-methods-expert skill

Overview

This skill is a numerical methods expert for computational mathematics focused on numerical solutions, approximations, error analysis, and scientific computing. It bundles legacy guidance and a research workflow that must be refreshed before production use. The skill flags areas with low confidence and provides a roadmap to validate and upgrade its recommendations.

How this skill works

The skill inspects and summarizes numerical algorithms, stability and convergence properties, and error estimation techniques across common problem classes (ODEs, PDEs, linear systems, optimization, interpolation). It guides users through a research checklist to capture current sources, document known gaps, and verify topic modules before relying on them. Outputs include algorithm comparisons, implementation notes in Python, and suggested validation tests.

When to use it

• When you need a compact expert overview of numerical algorithms and trade-offs for a specific problem
• Before implementing or selecting a solver when you need guidance on stability, convergence, and error control
• When auditing existing scientific code for numerical accuracy or sensitivity to parameters
• When assembling a validation plan or test matrix for numerical software
• When preparing to update or extend numerical method modules and needing a structured research workflow

Best practices

• Treat the skill as a draft: verify recommendations against up-to-date primary literature or textbooks
• Run small-scale numerical experiments to confirm stability and error behavior before scaling up
• Document all external sources and results in the provided research log for traceability
• Prefer reproducible examples in Python with explicit tolerances, mesh refinement, and convergence plots
• Flag and resolve known gaps before using recommendations in production code

Example use cases

• Compare implicit vs explicit time integrators for stiff ODEs and produce test cases demonstrating accuracy and cost
• Select preconditioners and solvers for large sparse linear systems arising from finite-element discretization
• Design mesh refinement and error estimators for a Poisson solver and validate convergence rates
• Estimate rounding and discretization error for a numerical integration routine and recommend mitigations
• Create a reproducible experiment notebook that documents algorithm variants, parameters, and outcomes

FAQ