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

✍ Scribed by S.C. Jardin; G. Bateman; G.W. Hammett; L.P. Ku

• Publisher: Elsevier Science
• Year: 2008
• Tongue: English
• Weight: 843 KB
• Volume: 227
• Category: Article
• ISSN: 0021-9991
• DOI: 10.1016/j.jcp.2008.06.032

✦ Synopsis

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 , 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 . 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: