This chapter introduces numerical methods, including 1) Finite Difference Approach, 2) Methods of characteristics (Eulerian-Lagrangian), and 3) Finite Element Approach for solving the ADE applicable to multidimensional, variable velocity, irregular boundary, and initial conditions. However, only one- and two-dimension examples are illustrated for convenience. Once the algorithms are understood, they can be expanded to other situations with ease.

In this chapter we study finite difference schemes for parabolic partial differential equations. The notions of conditional and unconditional stability, and CFL condition are introduced to analyze the classical schemes for the heat equation. Different techniques, like maximum principles and energy arguments are presented to obtain stability in different norms. Then, we turn to the study of the pure initial value problem, the grand goal being to discuss the von Neumann stability analysis. To accomplish this we introduce the notions of Fourier-Z transform of grid functions and the symbol of a finite difference scheme. This allows us to state the von Neumann stability condition and prove that it is necessary and sufficient for stability. These notions are also used to present a covergence analysis that is somewhat different than the one presented in previous sections.

In this chapter we develop a numerical model of the transport of chemical mass within flowing groundwater. Starting with a derivation of the advection–dispersion equation, we consider how the equation may be cast in discrete form as a finite difference equation. We further discuss the numerical stability of the calculation procedure and the numerical dispersion inherent in the discrete solution, concluding with a worked example.