Stability of Crank-Nicholson for variable diffusivity

September 7, 2007

Title: Stability analysis of the Crank-Nicholson method for variable coefficient diffusion equation

Author: Charles Tadjeran

Source: Communications in Numerical Methods in Engineering, Vol 23, Issue 1, pp. 29-34, 2006.

Abstract:
The Crank-Nicholson method is a widely used method to obtain numerical approximations to the diffusion equation due to its accuracy and unconditional stability.

When the diffusion coefficient is not a constant, the general approach is to obtain a discretization for the PDE in the same manner as the case for constant coefficients. In this paper, we show that the manner of this discretization may impact the stability of the resulting method and could lead to instability of the numerical solution. It is shown that the classical Crank-Nicholson method will fail to be unconditionally stable if the diffusion coefficient is computed at the time gridpoints instead of at the midpoints of the temporal subinterval. A numerical example is presented and compared with the exact analytical solution to examine its divergence.

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