Solve the characteristic equations numerically to solve a PDE

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

• Your question is unclear to me and I'm not sure what you're asking. Do you want to know how to finite difference the PDE? – Mattos May 9 at 11:21
• no I propose an alternative method to solve this PDE other than finite differences. I am asking what would be the way to solve numerically this pde using a numerical integration of the characteristic equations – Smilia May 9 at 12:11

You can indeed implement the proposed method, but you will need to choose a finite number of values of $$x_0$$, e.g. $$2K+1$$ values $$x_0 = X_0 + k\Delta x_0$$ for $$k = -K,\dots, K$$ and $$\Delta x_0 >0$$. However, this will not give the solution at any $$(x,t)$$. To get as close as possible to some arbitrary $$(x,t)$$, you may need to modify the chosen values of $$x_0$$. To do so automatically, one may optimize the parameter $$X_0$$ in the case of $$2K+1=1$$ values of $$x_0$$, e.g. by using dichotomy or fixed-point iterations.