On the application of spline functions and TSRK methods to systems of ODEs with maxima

Z. Bartoszewski, Z. Jackiewicz, T.Jankowski

Gdansk University of Technology, ul. Narutowicza 11/12, 80 952 Gdansk, Poland

zbart@mif.pg.gda.pl

Contributed talk

In the talk we discuss an adaptation of the following two-step Runge-Kutta method of order five and stage order five with variable step size ([1,2])

$$\left\{ \begin{array}{rcl} Y^{[n]} & = & (u\otimes I_m)\t... ...]})+ (w^T\otimes I_m)F(Y^{[n]})\right), \end{array} \right.$$

$n=1,2,\ldots,N-1$,    where for      $x_0<x_1<\cdots <x_N, \quad x_N\geq X$,     $h_{n+1}=x_{n+1}-x_n$,     $\tilde{y}_0=y_0$,    and for $n=2,3,\ldots,N$

$$\left\{ \begin{array}{ccl} F(\tilde{Y}^{[n-1]}) & = & (\t... ...lde{w}\otimes I_m)h_{n+1}F(Y^{[n-1]}). \end{array} \right.$$

The adaptation of this method is applied to solving problems of the form

$$\left\{ \begin{array}{rcl} y'(t)&=&f(t,y(t),\max_{s\in [\... ...T],\\ y(t)&=&\psi(t),\quad t\in [-h,0]. \end{array} \right.$$

The dense output obtained by applying interpolating cubic splines which use the approximations of the solution in two neighboring grid points and four stage values at the points between them allows to find effectively approximations to the sought maxima. Numerical experiments with one and two dimensional examples confirm the effectiveness of the proposed method.