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We consider an initial and Dirichlet boundary value problem for a semilinear,two dimensional heat equation over a rectangular domain. The problem isdiscretized in time by a version of the Relaxation Scheme proposed by C. Besse(C. R. Acad. Sci. Paris S\'er. I, vol. 326 (1998)) for the nonlinearSchr\"odinger equation and in space by a standard second order finitedifference method. The proposed method is unconditionally well-posed and itsconvergence is established by proving an optimal second order error estimateallowing a mild mesh condition to hold.

A Relaxation/Finite Difference discretization of a 2D Semilinear Heat Equation over a rectangular domain