TY - JOUR
T1 - The Stability and Convergence of Computing Long-Time Behaviour
AU - Wu , Hai-Jun
AU - Li , Rong-Hua
JO - Journal of Computational Mathematics
VL - 4
SP - 397
EP - 418
PY - 1999
DA - 1999/08
SN - 17
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/jcm/9111.html
KW - Stability, Compatibility, Convergence, Reaction-diffusion equation, Long-time error estimates.
AB - <p style="text-align: justify;">The object of this paper is to establish the relation between stability and convergence of the numerical methods for the evolution equation $u_t-Au-f(u)=g(t)$ on Banach space $V$, and to prove the long-time error estimates for the approximation solutions. At first, we give the definition of long-time stability, and then prove the fact that stability and compatibility imply the uniform convergence on the infinite time region. Thus, we establish a general frame in order to prove the long-time convergence. This frame includes finite element methods and finite difference methods of the evolution equations, especially the semilinear parabolic and hyperbolic partial differential equations.&nbsp;As applications of these results we prove
the estimates obtained by Larsson [5] and Sanz-serna and Stuart [6].</p>