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 S{\'e}r. I, vol. 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 $L_t^{\infty}(H_x^2)-$norm at the time-nodes and in the discrete $L_t^{\infty}(H_x^1)-$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.