Many computational fluids problems are described by nonlinear parabolic partial differential equations. These equations generally involve advection (transport) and a small diffusion term, and in some cases, chemical reactions. In almost all cases they must be solved numerically, which means approximating steep fronts, and handling time-scale effects caused by the advective and reactive processes. We present a time-splitting algorithm for solving such parabolic problems in one space dimension. This algorithm, referred to as the Godunov-mixed method, involves splitting the differential equation into its advective, diffusive, and reactive components, and solving each piece sequentially. Advection is approximated by a Godunov-type procedure, and diffusion by a mixed finite element method. Reactions split into an ordinary differential equation, which is handled by integration in time. The particular scheme presented here combines the higher-order Godunov MUSCL algorithm with the lowest-order mixed method. This splitting approach is capable of resolving steep fronts and handling the time-scale effects caused by rapid advection and instantaneous reactions. The scheme as applied to various boundary value problems satisfies maximum principles. The boundary conditions considered include Dirichlet, Neumann and mixed boundary conditions. These maximum principles mimic discretely the classical maximum principles satisfied by the true solution. The major results of this thesis are discrete L$\sp\infty$(L$\sp2$) and L$\sp\infty$(L$\sp1$) error estimates for the method assuming various combinations of the boundary conditions mentioned above. These estimates show that the scheme is essentially first-order in space and time in both norms; however, in the L$\sp1$ estimates, one sees a much weaker dependence on the lower bound of the diffusion coefficient than is usually derived in standard energy estimates. All of these estimates hold for uniform and non-uniform grid. Error estimates for a lower-order Godunov-mixed method for a fully nonlinear advection-diffusion-reaction problem are also considered. First-order estimates in L$\sp1$ are derived for this problem.