I need to solve such systems of PDEs numerically and I'm wondering what the "standard" method (if there is one) is. The systems I'm interested in have the form:
$\frac{\partial F_i}{\partial t} + \lambda_i \frac{\partial F_i}{\partial x} = c(t,x,\mathbf{F})$
for $i = 1, \ldots, n$ and $\mathbf{F}(t,x) = (F_1(t,x),\ldots,F_n(t,x))$. The dependence of $c(t,x,\mathbf{F})$ on $\mathbf{F}$ (as well as on $t$ and $x$) may be non-linear. Initial conditions are given for $\mathbf{F}(0,x)$.
I'm fairly new to PDEs and what I've been doing is using some version of the method of characteristics, that is, I've been defining functions $G_i(a,b) := F_i(a,b+\lambda_i a)$, so that, holding $b$ fixed, I obtain coupled ordinary differential equations describing the evolution of the functions $G_i(\cdot,b)$. Fixing a grid in both dimensions, these can then be solved simultaneously in $a$, potentially interpolating between $b$ values on the grid where necessary. I think this approach is referred to as the "method of specified intervals". I'm not really sure if it's the best approach to be adopting... I don't know anything about its stability/convergence properties. Are there any good references for this?
Another approach I'm aware of is finite differencing, for example the Lax--Wendroff method. However, all of the books I have access to seem to suggest that it is not recommended in the non-linear case. Is this true? If so, what are my options? Again, does anyone have any appropriate references?
Many thanks!