Parallel two-step ROW-methods for stiff delay differential equations
The authors investigate the GP-stability and GPL-stability of numerical methods for delay differential equations, \[ \left\{ \begin{matrix} y'(x)&=& f(x,y(x),y(x-\tau)), \quad x\geq 0\\ y(x)&=&g(x), \quad -\tau \leq x \leq 0. \end{matrix} \right. \] It is proved that a given parallel two-step ROW-method (PTSROW method) obtained by a Hout's interpolation technique from a method in the class of linearly-implicit methods obtained by \textit{H. Podhaisky, B. A. Schmitt} and \textit{R. Weiner} [Appl. Numer. Math. 42, No. 1--3, 381--395 (2002; Zbl 1005.65073)] is GP-stable or GPL-stable only if the one on the class is A-stable or L-stable respectively. Also, by means of tree theory and B-series the order of consistency conditions and the order of stage-order conditions for the PTSROW method is analyzed. The paper ends with the construction of two classes of stiffly-accurate PTSROW methods, also of two classes of real-time PTSROW methods, and with some numerical experiments implemented in parallel MPI C language to test accuracy and speed of sequential versus parallel implementation. The first example considers a stiff nonlinear ordinary differential equation, while the second is applied to a stiff linear delay differential equation.
scientific article; zbMATH DE number 5574497
scientific article