EPODE

Method properties

Analyzing the data inputs, we can classify the method and the problem class for which it can works. Most important for the computational process are the following properties:

the method classification as explicit or implicit one (needed a nonlinear equation solving procedure);
the method classification as multistep one or onestep;
the method classification as onestage or multistage one (intermediate values are calculated, but not stored for the next computations);
the method classification as onederivative or multiderivative one (usage only of f or also of the Jacobian);
the method order. Let Y(t₁) be a vector with the exact solution values approximated by Y₁. If Y₁-Y(t₁)=O(h₀^p+1), the method order is p. The method error Y₁-Y(t₁) produced at each step is accumulated in the global error (which also includes the truncation errors) at the final step. The control of the global error means first of all the control of the method error at each step;
the error constant. This value distinguish methods with the same order. Best are those with the lowest constant error. The error constant is the value c^* from Y₁-Y(t₁)=c^*h₀^p+1+ O(h₀^p+2).
the absolute stability region. The boundary of this stability region has a great influence on the boundless of the method step h. The absolute stability region is defined as follows:
A={hlin C | the method applied to the test equation y'=l y, with constant stepsize h, produces an approximation y_n goes to 0, when n goes to infinity}
For a specific problem, the stepsize must be selected so that hl_i(t_n)in A, where l_i(t_n) are the eigenvalues of the Jacobian matrix J_n=(d f/d y)(y_n)@ (d f/d y)(y(t_n)). If A is a bounded region in the complex plane, and if l_i(t_n) are very large, it is possible that the stepsize to be very small relative to the integration interval. Therefore, methods with a bounded A are not indicated to solve stiff problems. There are some method for which A is unbounded. In order to estimate the maximum allowed stepsize, we must approximate the boundless of A;
the time necessary for apply one method step to the test problem. This time can offer an estimation of the time necessary to apply the method to a specific problem, in order to obtain an approximative solution.