2.3.15 Consistency  stability  convergence

Consistency:
A numerical scheme is
consistent if its
discrete operator (with finite differences) converges towards the
continuous operator (with derivatives) of the PDE for
(vanishing truncation error).

Stability:
``Noise'' (from initial conditions, roundoff errors,...) does not grow.

Convergence:
The solution of the numerical scheme converges towards the real solution of the PDE
for
.
Lax's equivalence theorem:
``Given a properly posed initial value problem and a finitedifference approximation to it
that satisfies the consistency condition,
stability is the necessary and sufficient condition for convergence.''

Consistency:
discrete operator
PDE operator

Stability:
discrete operator does not amplify ``noise''

Convergence:
Numerical solution
real solution
Lax's equivalence theorem:
consistency
stability
convergence