A difference scheme with the optimal weight for the diffusion-convection equation

A difference scheme with weights for a homogeneous spatially one-dimensional diffusion-convection equation is studied. An analysis of the approximation error for the difference scheme as a time step function is performed on the basis of the expansion of the solution and approximation error in a trigonometric basis. An algorithm is proposed to find the optimal weight value that ensures the minimum approximation error of the solution to an initial boundary value problem for given values of the time grid steps. A better accuracy of the constructed scheme with the optimal weight compared to the explicit scheme as well as the efficiency of the algorithm for finding the optimal weight value is shown using a test problem.