Numerically solving a transport equation

Question

I would like to solve this transport PDE numerically :

$$ \partial_t f + v(f) \partial_x f = 0 $$

What I would want to do is "freeze" the velocity $v$ and solve a classical transport equation by inverting the characteristics. For instance I would - in a fixed point fashion - first freeze $v$ with $f_0$ (first guess), then compute $f_1$, freeze $v$ with $f_1$, compute $f_2$ and so on and so forth. The question is : what condition on $v$ is required for this algorithm to converge ?

Answer 1

Follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:

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

$\dfrac{df}{ds}=0$ , letting $f(0)=f_0$ , we have $f=f_0$

$\dfrac{dx}{ds}=v(f)=v(f_0)$ , letting $x(0)=F(f_0)$ , we have $x=F(f_0)+v(f_0)s=F(f)+v(f)t$

Answer 2

If you want to use a finite difference method then use an upwind method.

If you prefer finite elements the a SUPG (streamline upwind Petrov Galerkin) approach is your best bet.

Normal finite difference and finite element methods are unstable for this class of first order hyperbolic problems.