IMI/Publicaţii/BASM/Ediţii/BASM n.2 (27), 1998/

A computation of the solutions for semilinear parabolic problems using implicit Runge-Kutta methods (Radau type). (English)

Authors: Axelsson O., Belousova N.


The following problem of the type $\frac{\partial u}{\partial t} =\Delta u + \bold{b} \cdot \nabla u + \tilde{f}(u), \ t>0, \break x \in \Omega \subset \bold{R}^d, \ d=1,2,3 $ with standard boundary conditions, the initial condition $u(0,x) = u_0(x)$ and monotone operator $-f$ is considered. It is assumed that the solution has a transient initial phase. The purpose of this paper is to investigate the Implicit Runge-Kutta (IRK) methods (Radau type). It is shown that there doesn't exist a right scaling matrix $D=diag(d_1,d_2,...,d_s)$ such that an error estimate in the form $\|e(t+\tau)\|_G\leq \|e(t)\|_G ~ + \tau \,\, \|R_s(t,w)\|$ can hold. The stability behaviour of the IRK methods (Radau type) is analysed. Further it is proved that s-stage IRK methods (Radau type) are B-stable. As a test-example the population ecology model and a number of numerical experiments are considered.

Faculty of Mathematics and Informatics
Nijmegen Catholic University
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Institute of Mathematics Academy of Sciences of Moldova
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