$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Consistency, stability, and convergence
- Truncation error measures the residual in the difference equations.
The scheme is consistent if the truncation error goes to 0
as \( \Delta t\rightarrow 0 \). Importance: the difference equations
approaches the differential equation as \( \Delta t\rightarrow 0 \).
- Stability means that the numerical solution exhibits the same
qualitative properties as the exact solution. Here: monotone,
decaying function.
- Convergence implies that the true (global) error
\( e^n =\uex(t_n)-u^n\rightarrow 0 \) as \( \Delta t\rightarrow 0 \).
This is really what we want!
The Lax equivalence theorem for linear differential equations:
consistency + stability is equivalent with convergence.
(Consistency and stability is in most problems
much easier to establish than
convergence.)
