A basic phenomenon modeled computationally is tracer transport in a flow field, such as in porous medium simulation. We analyze the stability and convergence of a fully conservative characteristic method, the Volume Corrected Characteristics-Mixed Method [4] (VCCMM) applied to advection of a dilute tracer in an incompressible flow. Numerical tests for the optimal convergence rate match the results of our theoretical proof. We avoid the CFL constraint on the time step size and obtain a higher order convergence rate compared with Godunov's method. We describe the implementation of the VCCMM, where we feature and define a polyline class for the volume computation of trace-back regions. Some numerical examples show that large time steps can be used in practice, no overshoot or undershoot arises in the solution, and less numerical diffusion is produced compared with Godunov's method. An application to a nuclear waste disposal problem is also presented, where we simulate the processes of advection, reaction, and diffusion of radioactive elements in a simplified far field model. Finally, an extension of the VCCMM is developed for compressible flows, and a stability and convergence analysis is presented. / text