Abstract: In this paper, a Euler-Taylor-Galerkin algorism for constant coefficient one dimension convective problem is introduced. To obtain the accurate temporal differencing, the method employs forward-time Taylor series expansions including time derivatives of second and third order which are evaluated from the governing partial differential equation. This yields a general time discretized equation which is successively discretized in space by means of the standard Bubnov-Galerkin finite element method. The technique is illustrated first in one dimension. With linear elements and Euler time stepping , several interesting relations with standard Galerkin and recently developed Petrov-Galerkin methods emerge and new Taylor-Galerkin schemes are found to exhibit particularly high phase accuracy with minimal numerical damping.