The stability of explicit Euler time‐integration for certain finite difference approximations of the multi‐dimensional advection–diffusion equation

A comprehensive study is presented regarding the numerical stability of the simple and common forward Euler explicit integration technique combined with some common finite difference spatial discretizations applied to the advection‐diffusion equation. One‐dimensional results are obtained using both the matrix method (for several boundary conditions) and the classical von Neumann method of stability analysis and arguments presented showing that the latter is generally to be preferred, regardless of the type of boundary conditions. The less‐well‐known Godunov‐Ryabenkii theory is also applied for a particular (Robin) boundary condition. After verifying portions of the one‐dimensional theory with some numerical results, the stabilities of the two‐ and three‐dimensional equations are addressed using the von Neumann method and results presented in the form of a new stability theorem. Extension of a useful scheme from one dimension, where the pure advection limit is known variously as Leith's method or a Lax‐Wendroff method, to many dimensions via finite elements is also addressed and some stability results presented.