Do you know any introductory references about integrating numerically the characteristic equations and deduce the general solutions of a pde?
More precisely I am interested in the transport equation $$ u_t + a(t,x)\, u_x = f(t,x)\\ u(0,x) = g(x) $$ where $g$ is a given function.
The characteristic equations are given by: $$ \dot x(t) = a(t,x(t)),\quad x(0)=x_0\\ \dot y(t) = f(t,x(t)),\quad y(0)=g(x_0) $$
If I solve this system numerically then I have to fix $x_0$ but this $x_0$ should move along the characteristic curve.... This amounts to solve (infinitely) many times these odes for different $x_0$ ? But more importantly, the problem is that numerically, I can't (in the general case) find the fonction: $x_0(t,x(t))$ that is I know $x(t)$ given $x_0$ but I can't find numerically $x_0$ in terms of $t$ and $x(t)$.
I found this book : http://library1.org/_ads/0AE6470890D1F950C8FB0DF64CBF914E chapter 4 but it integrates only the solution $u$ along characteristics. In that case, to obtain the graph of the solution $(x,y,u(x,y))$ we need to solve the characteristic equations where $x_0$ varies in some interval... so a priori to solve infinitely many ode