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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\\{\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\\{\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]\\{\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]\\{\nabla^2 \phi} &= {-e \over \epsilon} (n_h + n_c -n_e -n_a)\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?

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These PDEs are typical diffusion-drift PDEs that arise in a huge number of fields (including my field, which is electrochemistry) and are routinely solved using any numerical method of your choice, e.g. finite difference method and finite element method, in both open-source and commercial software. Nonlinearities are generally not an issue in my experience because we have access to very sophisticated time integrators and adaptive meshing.

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    $\begingroup$ In our experience, achieving convergence in this PDE is extremely complex when your boundary conditions assume that the flux of electrons ($j_e$) and holes ($j_h$) at both sides of the junction is also a function of the electric field (E) at such point ( i.e. Fowler-Nordheim tunneling process). In fact, many contributions have stated their inability to process and converge such a complicated system even for a steady state solution. (deMello, J. C., Phys. Rev. B, 2002, 66, 235210). $\endgroup$ – SalvaCardona May 4 at 13:24
  • $\begingroup$ @SalvaCardona Interesting, I will take a look later. $\endgroup$ – edwinksl May 4 at 13:42

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