A nonlinear relaxation formulation of the p-curl problem modelling high-temperature superconductors: A modified Yee's scheme

We propose a new tool for the study of the parabolic p-curl problem modelling the magnetic field in a high-temperature superconductor (HTS), a problem involving an operator $\nabla \times \left(\right|\nabla \times H{|}^{p-2}\nabla \times H)$ analogous to the p-Laplacian. This so-called nonlinear p-curl problem is computationally expensive in three space dimensions because sharp fronts in the magnetic field develop near the surface of HTS. The proposed technique is a relaxation model of the p-curl problem which is shown to provide a consistent and stable approximation according to first, an inner/outer layer analysis and secondly, according to a time scale expansion. The relaxation model leads to monotone approximations of the front because of the presence of anisotropic diffusion. Most importantly, it can be used to develop new schemes that inherit the properties of the relaxation model. In order to demonstrate the relevance of the relaxation model, this paper applies the approximation to the construction of a second-order nonlinear finite-difference time-domain (FDTD) method, similar to Yee's scheme. It is shown that the natural discretization of the p-curl inspired by Yee's scheme leads to an unstable scheme, while a discretization based on the relaxation model is stable. We propose two numerical discretizations that are respectively first and second order in time, but both second-order in space. The linear and nonlinear stability of the schemes is studied and it is demonstrated that the schemes are both discrete divergence free. We verify the numerical schemes using manufactured solutions and Mayergoyz's moving-front solution in both one and two space dimensions, as well as on a magnetization of a typical HTS.