Numerical integration of differential equations with processors

Sergio Blanes, Fernando Casas, Ander Murua

Departament de Matemàtiques, Universitat Jaume I, 12071-Castellón, Spain.

sblanes@mat.uji.es

http://www3.uji.es/~sblanes

Contributed talk

Let us denote by $x(t)=\varphi_t(x(0))$ the solution of a differential equation. A method, $\psi_h$ is of order $q$ if

$$\psi_h = \varphi_h + {\cal O}(h^{q+1}),$$ (1)

and it is of effective order $p$ if there exist a map $\pi_h$ such that [4]

$$\pi_h \circ \psi_h^n \circ \pi_h^{-1} = \varphi_h + {\cal O}(h^{p+1}).$$ (2)

Usually, $\psi_h$ is referred as the kernel and $\pi_h$ as the processor or corrector. Here, $\pi_h$ can be used to solve many of the order conditions as well as to cancel higher order non-correctable error terms [2,1]. This technique allow us to obtain accurate results at the cost of $\psi_h$ if the output is not required frequently. We show that, if the non-correctable error terms are small [5], the method will be very efficient, even if the output is required frequently.

The same accuracy as with (2), applied for $n$ steps, can essentially be obtained with

$$\hat{\pi}_h\circ \psi_h^n\circ \pi_h^{-1}(x(0))$$ (3)

where $\pi_h^{-1}$ is usually expensive since it has to satisfy many conditions, but it is computed only once, and $\hat{\pi}_h$ is a cheap approximation to $\pi_h$, obtained from the internal stages in the computation of $\psi_h$ so, it is virtually cost-free [3].