Concept / Scientific Computing
A computed answer can be precise and still be wrong.
Numerical stability concerns how errors behave as an algorithm proceeds. Near difficult boundaries, small representation or approximation errors can grow until the computed state no longer tracks the intended mathematics.
Four questions before trusting a result
| Layer | Question |
|---|---|
| Model | Do the equations and assumptions represent the phenomenon at the scale being studied? |
| Conditioning | Does a small change in valid input produce a large change in the exact answer? |
| Discretization | How does replacing a continuous object with a finite approximation change the result? |
| Algorithm | Does the computational procedure amplify rounding and truncation error? |
Conditioning is not stability
An ill-conditioned problem is inherently sensitive: nearby inputs can have very different exact outputs. An unstable algorithm adds avoidable sensitivity through its procedure. A stable method cannot remove the underlying conditioning of the problem, but it should avoid introducing much more error than that conditioning requires.
Why smaller steps are not always better
Reducing a finite-difference step can decrease truncation error, but eventually subtraction of nearly equal floating-point values can amplify roundoff. Reliable computation therefore studies an error regime rather than assuming that the smallest representable step is the most accurate.
As h shrinks, the approximation improves in exact arithmetic up to the order of the method. In floating-point arithmetic, cancellation in the numerator and division by a tiny h can eventually make the estimate worse.
Boundary-specific risks
- Sampling never reaches an excluded singular endpoint, so a cutoff must be explicit.
- A coarse mesh can hide a boundary layer or steep gradient.
- Clipping or chart scaling can make divergent behavior look bounded.
- Regularization can improve computation while changing the original problem.
- A stable-looking finite run does not establish convergence of an infinite limiting process.
Sources
- MIT OpenCourseWare, Introduction to Numerical Methods, Week 1, on accuracy, roundoff, and approximation.
- MIT OpenCourseWare, Numerical Fluid Mechanics lecture notes, on error types, conditioning, error propagation, and stability.