Is there a general method for determining the domain of dependence of (higher-order) PDEs? I would be plenty happy with a reference to a paper or textbook I could look at; despite much reading around, I couldn't find this problem addressed for anything besides first- and second-order PDEs.

If there's a simple answer to this question, then I probably don't need to write anything else. But for completeness, I'll provide background on the subject and on my particular interest.

## Domain of dependence

The **domain of dependence** of point $(\vec{x}_0,t_0)$ on the solution $U(\vec{x},t)$ of a hyperbolic$^\dagger$ PDE is the subset of the initial conditions which uniquely determine the value $U(\vec{x}_0,t_0)$. For the Advection equation

$U_t + c U_x = 0$

with initial conditions $U(x,0)=f(x)$, the domain of dependence of $(x,t)$ is just the point $x-c t$. For the wave equation

$U_{tt} - c^2 U_{xx} = 0$

the domain of dependence of $(x,t)$ is the line $[x-ct,x+ct]$. In every book I've read, the domain of dependence is only given for PDEs which have been explicitly solved (like these two examples) and no domain is given for a higher-order PDE without explict solutions.

## Background for my problem

I'm trying to simulate the evolution of the Wigner function (a pseudo probability distribution over phase space) for a point particle moving in a chaotic potential. The PDE governing the Wigner function $W(x,p,t)$ can be approximated as

$\partial_t W = -\frac{p}{m} \partial_x W + V^\prime (x) \partial_p W - \frac{\hbar^2}{24} V^{\prime \prime \prime} (x) \partial_p^3 W$

or, in the dimensionless PDE notation,

$W_t = -p W_x + f(x) W_p - f^{\prime \prime}(x) W_{ppp} .$

It appears that an instability in my simulation is due to my time steps not satisfying the CFL condition and I'm trying to rigorously derive the CFL condition for this PDE.

This may not be possible, in which case I'll use the CFL condition under the approximation that $f''(x)=0$.

$^\dagger$ Here I am using "Hyperbolic" in the sense that the Cauchy problem is well defined *without* restricting to second-order PDEs.