Stability of the standard Crank–Nicolson–Galerkin scheme applied to the diffusion-convection equation: Some new insights

A stability analysis of the classical Crank–Nicolson–Galerkin (CNG) scheme applied to the one-dimensional solute transport equation is proposed on the basis of two fairly different approaches. Using a space-time eigenvalue analysis, it is shown, at least for subsurface hydrology applications, that the CNG scheme is theoretically stable under the condition <i>PeCr</i>≤2, where <i>Pe</i> and <i>Cr</i> are the mesh Péclet and Courant numbers. Then, to assess the computational stability of the scheme, the amplification matrix is constructed, and its norm is displayed in the (<i>Pe, Cr</i>) space. The results indicate that the norm of the amplification matrix is never smaller than unity and exhibits a hyperbolic nature analogous to the above theoretical condition.