B2 - convergence

the accuracy of a numerical method is determined by its proximity to the analytical solution
considering the analytical solution of an abstract problem in space and time, $y (x, t)$ , and its numerical counterpart ${\hat{y}}_{k}^{j}$
constructing a measure of error at a point in space at a given time:

σ_{C} \equiv lim_{Δ x, Δ t \to 0} | {\hat{y}}_{k}^{j} - y (x_{k}, t_{j}) | \to 0

if a finite difference method is consistent and stable, the method converges, satisfying the limit
stability and consistency imply that a numerical scheme converges
- stability: the method does not introduce false and growing oscillations; if it is linearly unstable, any small oscillation will grow without bound
- consistency: chosen discrete representation of derivatives gives a correct representation of analytical equation and the solution and the difference in the error is well defined everywhere