On 1D diffusion problems with a gradient-dependent diffusion coefficient

0021-9991/$ see front matter 2008 Elsevier Inc doi:10.1016/j.jcp.2008.06.032 * Corresponding author. Tel.: +1 609 243 2635; fa E-mail address: [email protected] (S.C. Jardin). In solving the 1D (flux surface averaged) transport equations for the temperatures, magnetic fields, and densities in the ‘‘evolving equilibrium” description of a tokamak [1], one increasingly encounters highly nonlinear thermal conductivity and diffusivity functions, such as GLF23 [2], that have a strong and non-analytic dependence on the temperature gradients. These arise from a subsidiary microstability based calculation in which the growth rates and hence transport coefficients are sensitive functions of these gradients [3]. When these nonlinear functions are interfaced with an existing transport framework that uses a standard implicit time advancement algorithm such as Crank-Nicolson or backward Euler [4], large non-physical oscillations can develop and, as a result, non-convergent solutions can occur. Here we describe a relatively simple modification to these implicit algorithms that cures this difficulty. To illustrate the method, we start with a simple diffusion equation in cylindrical polar coordinates: