Numerical Stability and Convergence
Stability keeps errors from growing during a computation; convergence ensures the answer approaches truth as resolution improves.
Two separate guarantees
Stability and convergence are distinct properties that a numerical method must have to be trustworthy. Stability means small errors — from round-off or data — do not grow uncontrollably as the computation proceeds. Convergence means that refining the mesh or timestep drives the numerical answer toward the true solution.
Instability in action
An unstable time-stepping scheme can take a tiny round-off error and double it every step until the result is meaningless. This often shows up as a solution that oscillates wildly or blows up to infinity, even though the underlying physics is perfectly well behaved.
The timestep condition
Explicit schemes for time-dependent problems are often stable only if the timestep is small enough relative to the mesh spacing — a condition tying resolution in time to resolution in space. Violate it and the computation diverges; respect it and it behaves. Knowing such conditions is basic to running a solver correctly.
Demonstrating convergence
- Solve the same problem at increasing resolution.
- Check that the answer changes by less and less.
- Confirm the error shrinks at the method’s expected order.
- Report the resolution at which the result is effectively settled.
Why it is verification
A convergence study is one of the most important verification steps for any physics simulation. A result reported without evidence that it is converged and stable is not yet a physics answer; it may simply be an artifact of the chosen resolution.