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
Best Answer
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.