A Modified Method of Characteristics (MMOC) combined with Galerkin finite elements has often been used in the past to solve the advection dominated advection-diffusion equation that arises in miscible displacement, transport of soluble contaminants in groundwater and the bioremediation of contaminated aquifers. In this method the hyperbolic part of the equation is treated with characteristics and the remaining elliptic equation is solved with Galerkin finite elements. The right-hand-side in the later procedure is obtained through numerical integration. We demonstrate here that the error associated with this numerical integration is substantial enough to cause large material balance errors and poor convergence, even in the absence of overshoot and undershoot, typical of the Galerkin procedure. One can reduce the error by taking many quadrature points but at the cost of large CPU time. Finer discretization in space a and in time does not guarantee a better solution when the right-hand-side is not computed exactly . An exact integration scheme is implemented in 1-D. This is conservative and obtains theoretical convergence in the absence of overshoot and undershoot.