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

Authors: Axelsson O., Belousova N.

### Abstract

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
Toernooiveld 1, 6525 ED Nijmegen, the Netherlands
Institute of Mathematics Academy of Sciences of Moldova
Academiei str. 5, MD-2028 Chishinau, Moldova