The Global Dynamics of Discrete Semilinear Parabolic Equations

A class of scalar semilinear parabolic equations possessing absorbing sets, a Lyapunov functional, and a global attractor are considered. The gradient structure of the problem implies that, provided all steady states are isolated, solutions approach a steady state as $t \to \infty $. The dynamical properties of various finite difference and finite element schemes for the equations are analysed. The existence of absorbing sets, bounded independently of the mesh size, is proved for the numerical methods. Discrete Lyapunov functions are constructed to show that, under appropriate conditions on the mesh parameters, numerical orbits approach steady state solutions as discrete time increases. However, it is shown that insufficient spatial resolution can introduce deceptively smooth spurious steady solutions and cause the stability properties of the true steady solutions to be incorrectly represented. Furthermore, it is also shown that the explicit Euler scheme introduces spurious solutions with period 2 in the timestep. As a result, the absorbing set is destroyed and there is initial data leading to blow up of the scheme, however small the mesh parameters are taken. To obtain stabilization to a steady state for this scheme, it is necessary to restrict the timestep in terms of the initial data and the space step. Implicit schemes are constructed for which absorbing sets and Lyapunov functions exist under restrictions on the timestep that are independent of initial data and of the space step; both one-step and multistep (BDF) methods are studied