# PDE solver for drift-diffusion with generation

I am trying to solve the drift-diffusion model with generation,

$$\frac{\partial N_e}{\partial t} = \alpha(x) \left| \Gamma_e (x,t) \right| - \frac{\partial \Gamma_e}{\partial x} (x,t)$$ $$\frac{\partial N_i}{\partial t} = \alpha(x) \left| \Gamma_e (x,t) \right| - \frac{\partial \Gamma_i}{\partial x} (x,t)$$

where

$$\Gamma_e (x,t) = -\mu_e(x) E(x) N_e(x,t) - D_e(x) \frac{\partial N_e}{\partial x} (x,t)$$ $$\Gamma_i (x,t) = \mu_i(x) E(x) N_i(x,t) - D_i(x) \frac{\partial N_i}{\partial x} (x,t)$$

I was looking at upwinding schemes but am not sure if I can use this method when there is some diffusion? If so how would I handle the $0^{\mathrm{th}}$ and $2^{\mathrm{nd}}$ order terms?

I am trying to use finite differencing.

• Just use upwind differences for the drift terms and centered differences for the diffusion terms. – David Ketcheson Jun 11 '16 at 11:58
• @DavidKetcheson please, consider converting this comment to an answer. Even though it is short, it certainly is an answer. And maybe the answer. – Anton Menshov Jun 10 '19 at 22:41