I am trying to simulate electronic devices with sharp interfaces such as bilayer organic solar cells and diodes under the high applied voltage regimes to study the physics of their breakdown. There are steep gradients of important quantities, such as charge concentrations and electric potential, at the interfaces which causes the Scharfetter-Gummel finite difference method to become unstable. I have found that reducing the mesh size does not significantly help. What are possible alternative methods which would be stable and accurate?
You can use finite element method to solve the differential equations which will make it possible for you to use nonuniform meshing. So in the regions of high gradients you can use finer mesh and in the regions of smaller gradients you can use coarser mesh. You can follow yourself the appropriate numerical solutions using such techniques.
Best wishes
Any time you have sharp gradients, shocks or discontinuous solutions, finite volume schemes are well suite for your problem. The building blocks for this kind of numerical schemes are the so called Riemann problems, that consist in the solution of the PDE with discontinuous initial conditions in adjacent numerical cells.
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