Numerical Methods
Numerical methods approximate the answers to math problems that have no closed-form solution, which describes most real physics.
Why we approximate
Most equations describing a fusion plasma or a structural component cannot be solved exactly with pen and paper. Numerical methods replace the continuous problem with a finite one a computer can solve, accepting a controlled error in exchange for an answer.
The main families
- Root finding and linear algebra: solve equations and systems of equations.
- Interpolation and quadrature: fill gaps between data and integrate functions.
- Ordinary and partial differential equation solvers: advance systems in time and space.
- Optimization: find inputs that minimize or maximize an objective.
Error and its sources
Numerical answers carry error from several sources: discretization error from replacing continuous quantities with finite ones, round-off error from finite-precision arithmetic, and truncation error from stopping an infinite process early. Good practice means knowing which dominates.
Convergence
A trustworthy method converges: as the mesh is refined or the timestep shrinks, the answer approaches the true solution at a predictable rate. Demonstrating convergence is a basic form of verification — a method that does not converge is not yet giving you a physics answer.
Stability
A method is stable if small errors do not grow without bound as the computation proceeds. An unstable scheme can turn a tiny round-off into nonsense. Choosing methods whose stability matches the problem is as important as choosing ones that are accurate.