I have a very basic question, and I hope some of you might be able to help me: In Fluid Dynamics, a common equation turning up time and time again is the so-called continuity equation:$$\frac{\partial}{\partial x}(T\frac{\partial h}{\partial x}) + R = S\frac{\partial h}{\partial t}$$Where $T$ is the transmissivity of the soil matrix, defined by the product $Kh$, where $h$ is the hydraulic head and $K$ is the hydraulic conductivity (a constant), $R$ a recharge constant, and $S$ the storage coefficient, another constant. In FD solutions for the unconfined case (where the transmissivity varies with $h$) this equation is often linearized by assuming that $T$ is constant. If you ignore this linearization and continue, you get:$$\frac{\partial}{\partial x}(Kh\frac{\partial h}{\partial x}) + R = S\frac{\partial h}{\partial t}$$Now I don't see a problem with solving this. Couldn't one just apply the chain rule of differentiation to get...$$\frac{\partial Kh}{\partial x}\frac{\partial h}{\partial x} +Kh\frac{\partial^2 h}{\partial x^2}+ R = S\frac{\partial h}{\partial t}$$... and then solve each of these terms by conventional centered differences and second order central differences? I can see that this kind of solution may be problematic for implicit schemes, where you will get a squared $h$ in the first term, but why is this not done in explicit schemes?
Linearization in Finite Difference Method: Why?
finite difference;nonlinear equations