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, al least for subsurface hydrology applications, that the CNG scheme is theoretically stable under the condition PeCr less-than-or-equal-to 2, where Pe and Cr are the mesh Peclet and Courant numbers. Then, to assess the computational stability of the scheme, the amplification matrix is constructed, and its norm is displayed in the (Pe, Cr) 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.