# Numerical Methods

> Numerical solvers. ODE/PDE, FEM, sparse linear algebra, stability analysis.

- Skill: `aselimc/numerical-methods` (Agent Skill)
- Install (CLI): `npx skillmds@latest add aselimc/numerical-methods`
- Raw SKILL.md: https://api.skillmd.com/api/skills/aselimc/numerical-methods/raw
- Safety review: pending
- Works with: Claude Code, Claude.ai, OpenAI Codex
- Category: Coding & Dev Tools
- Author: aselimc (https://skillmd.com/u/aselimc)
- Updated: 2026-09-10
- Page: https://skillmd.com/skills/aselimc/numerical-methods

---


# Numerical Methods

## ODE Solvers
```python
from scipy.integrate import solve_ivp
sol = solve_ivp(rhs, [0, 10], y0, method='RK45', rtol=1e-8)
# For stiff systems: method='Radau' or 'BDF'
```

## PDE / FEM
```python
# FEniCS
from fenics import *
mesh = UnitSquareMesh(32, 32)
V = FunctionSpace(mesh, 'P', 1)
u = TrialFunction(V); v = TestFunction(V)
a = dot(grad(u), grad(v)) * dx
L = f * v * dx
solve(a == L, u_h, bc)
```

## Sparse Linear Algebra
- Direct: LU/Cholesky for small-medium systems (<10^5 unknowns)
- Iterative: CG (SPD), GMRES (general) with AMG preconditioner for large systems
- Format: CSR for arithmetic, COO for construction

## Stability
- CFL condition for explicit time-stepping: `dt < dx / c`
- Stiffness detection: large ratio of eigenvalues -> use implicit solver

## Key Libraries
SciPy, JAX, FEniCS, PETSc, Julia DifferentialEquations.jl