# Numerical techniques for solving systems of first-order semi-linear hyperbolic PDEs in two variables

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!

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Apart from finite difference schemes, you could search for finite element/volume methods that can be applied to PDEs. They are a bit more complicated but provide a wider range of applications,especially for complicated domains. – chemeng Apr 27 '12 at 20:58
Thanks for your response. I don't think my domain is particularly complicated though. I would generally wish to solve the system on a rectangle, say $[0,T] \times [0,X]$. I'm not sure finite element methods give me anything then here over finite difference methods? – Jon Apr 27 '12 at 21:45
FEM and FDM are similar, but for non-linear problems they both require linerazation and Newton-method iterations for convergence.They can be very effective even for non-linear problems but and in case you want to develop your own codes, both of them can be very time-consuming. – chemeng Apr 27 '12 at 21:54