The term experimental order of convergence (English: experimental order of convergence , EOC ) is understood in the numerical analysis an estimate of the rate of convergence of a sequence. In order to calculate this, the limit value is assumed to be known.
Let be the already known limit of the sequence . The sequence converges with speed if there is a constant that satisfies the inequality
Fulfills. For the sake of simplicity, it is now assumed that the convergence can occur exactly
to be discribed. This formulation then also applies to the next term in the sequence
Division of the two equations yields
So it applies
where denotes the logarithm to the base . A conversion of the logarithm to the base results in the definition of .
Application: Numerical solutions of differential equations
Let be numerical solutions of a method that approximately solves ( partial ) differential equations . There are different values of a discretization parameter that describes the resolution of the discretization. In the one-dimensional case it is usually the length of the largest interval. In the higher-dimensional case, an analog measure is used for the fineness of the grating, for example the largest inscribed diameter in two dimensions . Let be the limit of the procedure for . Then the experimental order of convergence is dependent on and through
given. This case can be explained by an a priori error estimator of the form
motivate with constants . As before, simplicity is also used here
accepted. This applies to both discretization and . Dividing the two equations gives
.
So it applies
,
which after converting the logarithm to base 10 gives the formula for .
Relation to the true convergence order
No convergence can be demonstrated using the EOC, as this is assumed. If there is a convergent method, it cannot generally be said whether the actual convergence rate is overestimated or underestimated by the EOC.
Individual evidence
↑ G. Sacrifice, Numerical Mathematics for Beginners , 2001, p. 304.