The extended family of 2ISD-methods for differential stiff systems

E. I. Vasilev ; T. A. Vasilyeva ; M. N. Kiseleva

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 3 (2015), pp. 34-44.

Résumé

The new set of absolutely stable difference schemes for a numerical solution of ODEs stiff systems (1) is submitted: \begin{equation} \frac{d}{dt} u(t) = f (u), \quad t>0, \quad u(0) = u_0. \end{equation} The main feature of the set is the multi-implicit finite differences with the second derivatives of the desired solution. The expanded three-parameter $(\alpha, \beta, \gamma)$ set of 2ISD-schemes (2)–(3) is studied in more details in this paper. \begin{equation} \left\{ \begin{aligned} \frac{{\nu}_{n+1} - {\nu}_{n}}{ф} = \sum_{i=0}^2 (a_{1i} E + ф b_{1i} J_{n+i}) f_{n+i},\\ \frac{{\nu}_{n+2} - {\nu}_{n}}{2ф} = \sum_{i=0}^2 (a_{2i} E + ф b_{2i} J_{n+i}) f_{n+i},\end{aligned} \right. \end{equation} \begin{equation} (a_{ki})= \begin{pmatrix} \frac{101}{240}+3б-2в \frac{128}{240}+4в \frac{11}{240}-3б-2в\\ \frac{56}{240}-3г \frac{128}{240} \frac{56}{240}+3г \end{pmatrix} , \\ (b_{ki})= \begin{pmatrix} \frac{13}{240}+б-в -\frac{40}{240}+4б -\frac{3}{240}+б+в\\ \frac{8}{240}-г -4г -\frac{8}{240}-г \end{pmatrix} . \end{equation} At arbitrary $(\alpha, \beta, \gamma)$ parameters last difference equation in system (2) has 5th order of accuracy. We found that the set of absolutely stable 2ISD-schemes includes two families: the set of the $L$-stable schemes and the set of the schemes of heightened accuracy for linear problems. For example: at $б = 1/168, в = 0, г = 0$ we have $A$-stable scheme with 8th order of approximation, at $б = -53/5880, в = 1/148 , г = 6/315$ we have $L_1$-stable scheme with 7th order of approximation, at $б = -23/360, в = 1/60, г = 14/315$ we have $L_2$-stable scheme with 6th order of approximation. The testing of this difference schemes on linear and nonlinear problems with a different stiff power is conducted. The errors of a numerical solution as functions of integration step size are computed in numerical experiments. These results demonstrate high quality of stability and accuracy of the suggested 2ISD-schemes.

Détail

Keywords: $L$-stability, $A$-stability, stiff systems, implicit methods, multi-implicit methods, methods with second derivative.

Bibliographie

[1] E.\;I. Vasilyev, T.\;A. Vasilyeva, M.\;N. Kiseleva, “Multi-Implicit SD-Methods for Stiff Systems of Differential Equations”, Science Journal of Volgograd State University. Mathematics. Physics, 2012, no. 2 (17), 68–77 | DOI

[2] E.\;I. Vasilyev, T.\;A. Vasilyeva, M.\;N. Kiseleva, “$L$-Stability of Multi-Implicit Methods of 8th Order Differential Stiff Systems”, Science Journal of Volgograd State University. Mathematics. Physics, 2013, no. 1 (18), 70–83 | DOI

[3] K. Dekker, Ya. Verwer, Stability of Runge–Kutta Methods for Stiff Nonlinear Differential Equations, Mir Publ., M., 1988, 334 pp.

[4] T. Vasilyeva, E. Vasilev, “High Order Implicit Method for ODEs Stiff Systems”, Korean Journal of Computational Applied Mathematics, 8:1 (2001), 165–180 | MR | Zbl