Error Estimation of the Besse Relaxation Scheme for a Semilinear Heat Equation
It establishes rigorous error bounds for a numerical method applied to semilinear PDEs, filling a gap in the literature for the Besse relaxation scheme.
The paper provides the first optimal, second-order error estimate for a fully discrete approximation of a semilinear heat equation using the Besse relaxation scheme in time and central finite differences in space, achieving convergence in the discrete L∞(H1) norm.
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{é}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 $L_t^{\infty}(H_x^1)-$norm. It is the first time in the literature where an error estimate for fully discrete approximations based on the Besse relaxation scheme is provided.