Stability analysis of a class of nonlinear magnetic diffusion equations and its fully implicit scheme

We studied a class of nonlinear magnetic diffusion problems with step-function resistivity $\eta (e)$ in electromagnetically driven high-energy-density physics experiments. The stability of the nonlinear magnetic diffusion equation and its fully implicit scheme, based on the step-function resistivity approximation model ${\eta }_{\delta }(e)$ with smoothing, were studied. A rigorous theoretical analysis was established for the approximate model of one-dimensional continuous equations using Gronwall's theorem. Following this, the stability of the fully implicit scheme was proved using bootstrapping and other methods. The correctness of the theoretical proof was verified through one-dimensional numerical experiments.