I've been trying to write an explicit FDM for this nonlinear system of PDEs:
$$ \frac{\partial u}{\partial t} = f(u, v) + \frac{\partial ^2 u}{\partial {x}^2} - \frac{\partial}{\partial x} \left(u \frac{\partial v}{\partial x} \right) $$
$$ \frac{\partial v}{\partial t} = g(v) +\frac{\partial ^2 v}{\partial {x}^2} - \frac{\partial}{\partial x} \left(v \frac{\partial u}{\partial x} \right) $$
The linear terms are discretized in the standard manner. However, I am not quite sure how to handle the last terms in the equations. My approach is to first apply the product rule (so I get $-\left(\frac{\partial v}{\partial x}\frac{\partial u}{\partial x} + u \frac{\partial ^2 v}{\partial {x}^2}\right)$) and then use forward differences for each of the first derivatives. Is this approach correct? Are there any better ways to handle the last terms? Thank you very much in advance for the answers.
EDIT: I tried discretizing the equations as mentioned above, but found out that the term $u \frac{\partial ^2 v}{\partial {x}^2}$ causes a blow-up. If I disregard $u$ (which is discretized as $u_{j}^{m}$) in this term, the whole scheme appears to work. I've also tried discretizing $u$ as $u^{m}_{j - 1}$, but it didn't work.
With the Lagrange interpolants for $u$ and $v$ on $x = -h,0, h$ you can compute $(v u')'$ at the midpoint as \begin{equation} (vu')'(0) = (A u_{-1} + B u_{0} + C u_{1})/(4 h^2) + O(h^2) \end{equation} where $A := v_{-1} + 4 v_{0} - v_{1}$, $B = -8 v_0$, $C = -v_{-1} + 4 v_0 + v_{1}$, with $u_{k} := u(kh)$, etc.