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

  • $\begingroup$ 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? $\endgroup$ – Mattos May 9 at 11:21
  • $\begingroup$ 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 $\endgroup$ – 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.


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