numerical-stability

Numerical stability for PDEs is enforced by verifying CFL, Fourier, and von Neumann criteria against your time step, grid spacing, and scheme. This Skill analyzes those conditions across advection, diffusion, and reaction problems in 1D–3D, outputting stability verdicts and recommended time steps to prevent blow-up.

The CFL (Courant–Friedrichs–Lewy) condition is a stability requirement that constrains your time step relative to grid spacing and velocity. Explicit schemes violate it at their peril; this Skill checks CFL limits automatically and flags unsafe combinations before simulation.

Stiffness arises when eigenvalues or reaction rates span large ranges, forcing implicit schemes or very small time steps. This Skill detects stiffness via eigenvalue analysis and Jacobian inspection, helping you choose between explicit and implicit integrators.

Yes. This Skill supports stability checks for both explicit schemes (CFL, Fourier, von Neumann) and implicit schemes (matrix conditioning, stiffness detection), covering the full range of time discretizations for PDEs.

Provide grid spacing (dx), time step (dt), velocity, diffusivity, reaction rate, problem dimensions (1D–3D), and your discretization scheme. The Skill requires Python 3.8+ and NumPy; it outputs stability verdicts, diagnostics, and a safe recommended time step.

High matrix conditioning numbers indicate ill-posed systems where small input errors amplify into large solution errors. This Skill computes conditioning metrics to warn you when implicit solvers risk numerical inaccuracy or slow convergence.