Ion drift-diffusion under the effect of an external electric field is a phenomenon with a huge relevancy in light emitting devices, neuromorphic architectures and in general molecular electronics
Up to now, the most advanced approaches considered two steady-state operative models: Electrodynamical model (ED) and Electrochemical Doping model (ECD). But the key point to solve the resulting set of Partial differential equations is related to their non-linearity and time-dependence. These set include Drift and Diffusion equations for each type of carriers (ions, electrons and holes) and Poisson equation:
\begin{align}{\delta{n_e} \over \delta{t}} &= \mu_{e} \left[{kT \over e} \nabla^2n_e - n_e \nabla^2 \phi- (\nabla n_e)(\nabla \phi)\right] -k_{eh}n_en_h\tag{1}\\{\delta{n_h} \over \delta{t}} &= \mu_{h} \left[{kT \over e}\nabla^2n_h - n_h \nabla^2 \phi- (\nabla n_h)(\nabla \phi)\right] -k_{eh}n_en_h\tag{2}\\{\delta{n_a} \over \delta{t}} &= \mu_{a} \left[{kT \over e} \nabla^2n_a - n_a \nabla^2 \phi- (\nabla n_a)(\nabla \phi)\right]\tag{3}\\{\delta{n_c} \over \delta{t}} &= \mu_{c} \left[{kT \over e} \nabla^2n_c - n_c \nabla^2 \phi- (\nabla n_c)(\nabla \phi)\right]\tag{4}\\{\nabla^2 \phi} &= {-e \over \epsilon} (n_h + n_c -n_e -n_a)\tag{5}\end{align}
My question is related to programmable solving methods for highly non-linear PDEs. Even in the case that we have the algebraic expression of the PDEs, which methodologies are powerful enough to deal with such a complicated problem?
