ABSTRACTIn this article, a new unconditionally stable scheme, based on the Associated Hermite orthogonal functions combined with first-order upwind scheme (AH-FUS), is proposed for solving the convection–diffusion equation. To eliminate the time variable from the computations, the time derivatives are expanded by Hermite functions, and a Galerkin's temporal testing procedure is introduced to the expanded equation. A set of implicit difference equations are derived in AH domain under no convergent conditions, and the numerical results can be obtained by solving the expanded coefficients recursively. Two numerical examples were considered to verify the accuracy and the efficiency of the proposed scheme.