Convergence analysis of the semi-implicit Euler method for abstract evolution equations

The semi-implicit Euler discretization method is studied for abstract evolution equations in a Hilbert space H, like _ u = f(t; u; u) ; t 2 (0; T ] ; u(0) = u0, where f(t; ; v) is one-sided Lipschitz and R(I hf(t; ; v)) = H for h > 0 su ciently small, and f(t; u; ) is Lipschitz-continuous. Extension to Banach spaces is then pointed out. Ordinary and partial , di erential and integro-di erential equations or systems are included. For instance, _ u = A(t; u) + B(t; u), where A(t; ) is [strongly] dissipative and maximal, and B(t; ) is Lipschitz-continuous, fall into the previous class. The scheme is u n+1 = un+ t f((n+1) t; un+1; un), n = 0; 1; :::; N 1, where t := T=N . Two main computational advantages with respect to fully implicit methods are: (a) linearization of semilinear problems, and (b) decoupling of systems into lower-dimensional (stationary) subsystems, at each time step. In the latter case, parallelization becomes possible. A full error analysis is performed: consistency and stability are estabilished, and precise convergence estimates are obtained. Several applications, including reaction-di usion and hyperbolic systems, are nally given. 1 Permanent address: Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Universit a di Padova, Via Belzoni 7, 35131 Padova, Italy. Dipartimento di Matematica Pura e Applicata, Universit a di Padova, Via Belzoni 7, 35131 Padova, Italy.