First order quasilinear Partial Differential Equation methods What methods, not including the one of characteristic curves and surfaces, can I use to solve first-order quasilinear equations like this one?
$$u_x+u_y=u^2$$
Follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:

$\dfrac{dx}{dt}=1$ , letting $x(0)=0$ , we have $x=t$

$\dfrac{dy}{dt}=1$ , letting $y(0)=y_0$ , we have $y=t+y_0=x+y_0$

$\dfrac{du}{dt}=u^2$ , we have $u(x,y)=\dfrac{1}{f(y_0)-t}=\dfrac{1}{f(y-x)-x}$
Follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:

$\dfrac{dx}{dt}=1$ , letting $x(0)=0$ , we have $x=t$

$\dfrac{dy}{dt}=1$ , letting $y(0)=y_0$ , we have $y=t+y_0=x+y_0$

$\dfrac{du}{dt}=u^2$ , we have $u(x,y)=\dfrac{1}{f(y_0)-t}=\dfrac{1}{f(y-x)-x}$