I'm trying to solve the following equation$$\dfrac{\partial}{\partial x}\left(e^{au}\dfrac{\partial u}{\partial x}\right) = 0$$Of course, this equation can be solved analytically. I am trying to understand the way of working with exponential coefficients.For the approximation, I use the finite volume method which leads  to$$F_{i+1/2} - F_{i-1/2} = 0,\qquad F_{i-1/2} = \left.e^{au}\dfrac{\partial u}{\partial x}\right|_{i+1/2} $$For the further linearization, I'm trying the following two methods:simple iteration, where $e^u$ is calculated based on the previous iterationNewton's method (with the series expansion, dropping terms that are proportional to $\delta u^2$ and smaller)$$u^{k+1} = u^k + \delta u$$The first one worked well until some value of $a$, and the second one didn't work at all (I have not found the mistake yet)Is there any way to deal with exponential coefficient?  Will be glad for any advice. Thanks!
elliptic equation with exponential coefficient
nonlinear equations;numerical modelling;diffusion