EPODE

Stability and order

If the error propagation mechanism is under control, the numerical method is stable. In mathematical terms, this fact can be expressed in the following form. A method (formula, process, scheme) is stable if, for each differential equation with all solutions asymptotically stable ones, there are two values, K and h₀, so that

||y_n - z_n||<= K||y₀ - z₀||, " h: 0< h < h₀,

holds, where y_n and z_n are the values of the approximate solutions in the n^th point of the interval division, computed with the stepsize h and starting from the value y₀, respectively z₀.

The method is asymptotically stable if, for a given stepsize, the perturbations in the numerical solution do not grow from a step to another, i.e.

||y_n - z_n||<= ||y_n-1 - z_n-1||.

Generally, the difference between the approximate solution and the exact solution must vanish when the division points number becomes infinitely. That means that the approximation can be improved growing the number of the division points, i.e. increasing the computational effort. The consistency express this request in the theoretical case when the numerical algorithm can be applied without truncation errors.

The method order is the convergence order of the approximate values for stepsizes h which are going to 0 (n goes to infinity). The notion of accuracy supposes the consistency condition and a small error constant C, smallest than the stepsize h.