Error estimation of the Besse Relaxation Scheme for a semilinear heat equation

The solution to the initial and Dirichlet boundary value problem for a semilinear, one dimensional heat equation is approximated by a numerical method that combines the Besse Relaxation Scheme in time [ C. R. Acad. Sci. Paris Sér. I 326 (1998)] with a central finite difference method in space. A new, composite stability argument is developed, leading to an optimal, second-order error estimate in the discrete L t ∞ ( H x 2 )-norm at the time-nodes and in the discrete L t ∞ ( H x 1 )-norm at the intermediate time-nodes. It is the first time in the literature where the Besse Relaxation Scheme is applied and analysed in the context of parabolic equations.