Alternating-Direction Explicit (A.D.E.) finite-difference methods make use of two approximations that are implemented for computations proceeding in alternating directions, e.g., from left to right and from right to left, with each approximation being explicit in its respective direction of computation. Stable A.D.E. schemes for solving the linear parabolic partial differential equations that model heat diffusion are well-known, as are stable A.D.E. schemes for solving the first-order equations of fluid advection. Several of these are combined here to derive A.D.E. schemes for solving time-dependent advection-diffusion equations, and their stability characteristics are discussed. In each case, it is found that it is the advection term that limits the stability of the scheme. The most stable of the combinations presented comprises an unconditionally stable approximation for computations carried out in the direction of advection of the system, from left to right in this case, and a conditionally stable approximation for computations proceeding in the opposite direction. To illustrate the application of the methods and verify the stability conditions, they are applied to some quasi-linear one-dimensional advection-diffusion problems.

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Numerical Methods for Partial Differential Equations
School of Mathematics and Statistics

Campbell, L, & Yin, B. (B.). (2007). On the stability of alternating-direction explicit methods for advection-diffusion equations. Numerical Methods for Partial Differential Equations, 23(6), 1429–1444. doi:10.1002/num.20233