I have a 2D PDE of the form $$\nabla u \cdot \nabla F(x,y) = G(x,y)$$ with known functions $F$ and $G$ and unknown scalar function $u$, which I would like to solve numerically. $F$ and $G$ are determined from experimental data, and may have noise in them. Analytically, this is solved by the method of characteristics, and the obvious way to solve it numerically is to just integrate the characteristic equations. However, I'm not sure that this is the best approach: For one thing, it seems like if you integrate $u$ along a characteristic curve, after a long time the value will accumulate errors and diverge from the solution along neighboring curves, leading to a non-smooth solution. This might be exasperated by the noise in $F$ and $G$.
It seems like there should be a more robust approach (perhaps involving a small artificial diffusive term or something), but I haven't found anything in the usual references (Wikipedia, Numerical Recipes, etc.) on solving this sort of equation. I'm looking for concise techniques and/or references on robustly solving this sort of PDE.
Also, I'm no expert on this, so perhaps my worries about integrating the characteristics are unfounded, and that's a perfectly good way to solve this. I welcome someone to tell me that is so, if you know better.
Edit: Given the low uptake on this question, I'm starting a bounty, and I will clarify further what I am looking for:
Please provide references or other verifiable information on how to robustly solve (numerically) a linear, first order PDE. Preferably the solution should be known to work in the presence of noise on the coefficients. Conversely, if the method of characteristics is suitable for this purpose, please explain why.
The PDE rewrite as a 2D linear PDE of the form $$ au_x + b u_y = G $$ where $a=F_x$, $b=F_y$. A more versatile approach than integration along characteristics could be based on finite difference shemes or on finite volume schemes (see for instance the Lax-Friedrichs scheme). This becomes more obvious if $a\neq 0$, in which case $$ u_x + c u_y = R $$ with $c=b/a$ and $R=G/a$.