The Alternating Direction Explicit ( ADE ) Method

This chapter introduces the Alternating Direction Explicit (ADE) method and its applications to solving time-dependent Partial Differential Equations (PDE), in particular Black–Scholes-style equations. The method was invented by V.K. Saul'yev who applied it to solving time-dependent diffusion problems. In general, an effective way to learn ADE is to apply it to one-dimensional diffusion equations and then one-dimensional convection-diffusion equations. Ideally, domain transformation and Dirichlet-style boundary conditions should be used. The ADE family of methods differs in a number of subtle ways from the well-known finite difference schemes. First, they are explicit and stable, and the resulting discrete system of equations can be solved without the need to use matrix solvers. Second, some ADE variants are only conditionally consistent with a given PDE, which means that the step sizes in space and time must be related to each other.