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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.

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I don't believe that there is any general fact that would be useful to you. There are in fact very few useful general facts about PDE's and their domains of dependence (that's why you see so little about it in the textbooks). But you're at UCSB, so I recommend asking Gustavo Ponce if you could discuss your problem with hm or one of his students or colleagues. –  Deane Yang May 15 '10 at 17:32
Actaully, I'm at Los Alamos National Lab in New Mexico (UCSB on paper). Thanks, though. –  Jess Riedel May 16 '10 at 8:52
Out of curiosity, have you tried taking the Fourier transform of $p$-space? If you write $\hat{W}$ as the $p$-Fourier transform of $W$ (fixing $t,x$), then your equation is $-i \hat{W}_t = \hat{W}_{q,x} + f(x) q \hat{W} + f''(x) q^3 \hat{W}$. ($q$ is the Fourier conjugate to $p$). I am not too familiar with the CFL condition, but it seems that it encodes sensitivity to the high frequency components of the equation, which is exacerbated by the third derivative in $p$ you are taking. If your initial data is sufficiently smooth, the decay in Fourier space should be fast enough that... –  Willie Wong May 24 '10 at 0:27
...doing the computation in Fourier space with the $q^3$ weight may be okay and is more stable. This is just a random speculation though, as I am not an expert on numerical schemes. –  Willie Wong May 24 '10 at 0:28
I don't know much about this but maybe it is a good idea to look up literature on numerical solution of linear KdV equations. –  timur Sep 18 '11 at 16:32
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1 Answer

up vote 3 down vote accepted

While I am aware of some facts about domain of dependence properties for hyperbolic PDEs, I don't think most of them will be useful for you. The problem is that what you consider as hyperbolic (in your footnote) is too large of a class of equations for the notion to be useful: an illustration is the Heat equation. It is usually classified as a parabolic equation, but it does admit a well-posed initial value problem. So by your definition is hyperbolic. Now it is well known that the heat equation has infinite smoothing properties and infinite speed of propagation.

Now, in the special case of symmetric hyperbolic systems, even in higher orders, one can generally describe the domain of dependence by considering the characteristic cones (see, eg. http://eom.springer.de/c/c021610.htm ) of the system. Essentially the "fattest cone" gives you the maximum speed of propagation (which may depend quasilinearly on the solution) and integrating back this cone gets you the domain of dependence.

The domain of dependence properties are really closely associated to a priori energy estimates (see Courant and Hilbert, Methods of Mathematical Physics).

But I don't think this will solve your problem since I don't believe your question can be recast in a form in which such estimates are available.

In particular, looking directly at your equation, on the spatial side it has potentially infinite speed of propagation since the spatial propagation is essentially just a transport equation. So if $W(x,p,t)$ has non-compact support, then the spatial propagation can have arbitrary large speeds. So if your potential vanishes or if your initial data is homogeneous in momentum, your solution will have, as its spatial domain of dependence given by the largest and smallest momentum at which $W$ is supported.

Assuming $f'' = 0$, then you equation can be solved by the method of characteristics: $\partial_t W = v\cdot\nabla W$ where $v(x,p) = (p,f(x))$ is a vector field. The domain of dependence for this problem can be easily found by integrating the vector field independently of the function $W$. I don't know how to deal with your third order term.

Like I said, in general there are only two ways to study domain of dependence properties that are well established, the first is via explicit notion of the Green's function, the second is energy estimates.

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(1) Was I correct in thinking the definition of "hyperbolic PDE" was overloaded? Wikipedia uses the definition in my footnote, but many textbooks only use it to for second-order PDEs satisfying $b^2-4ac>0$. (2) Isn't impossible, then, to have numerical convergence if there is instantaneous propagation because the numerical domain of dependence is always finite? (3) Thanks very much for the thoughtful answer. –  Jess Riedel May 15 '10 at 23:31
(1) Yes. The ones in the textbooks are over-restrictive (and it only really applies to second order PDEs in 1+1 dimensions), but the Wikipedia definition is, in some sense too lax: it is rather hard to characterize an equation as hyperbolic by that way. More useful notions of hyperbolicity are the strict (Leray), symmetric (Lax and others), and regular (Christodoulou), which are algebraic conditions on the symbol. But all of these notions require that the total number of derivatives be balanced between the spatial and temporal directions, due to the use of Cauchy Kovalevskaya for the IVP. –  Willie Wong May 24 '10 at 0:11
(2) Not so. Numerical convergence can be had if you can show that far away bits contribute less, so you can forget about the errors coming from close to infinity and chop off. See again the heat equation. Numerical solutions for the heat equation is more or less stable because contributions from far away regions decays exponentially (see the explicit form for the heat kernel), even though it has infinite speed of propagation. –  Willie Wong May 24 '10 at 0:18
on (1): I generally agree with Willie's answer but you can rule out the heat equation by saying that strictly speaking it is not well posed unless you restrict the growth at infinity. For the constant coefficient case Garding (1951) proved that hyperbolicity in the sense of OP is equivalent to hyperbolicity in terms of the principal symbol. For more general equations there are no complete results which is why I generally agree with Willie. –  timur Sep 18 '11 at 14:48
a tiny addition to (2): infinite speed of propagation is in principle no problem as Willie explained, but it still forces you to choose very small time steps in explicit time integration methods. For this purpose the characteristic surface is not really a plane, but it acts more like a paraboloid. Implicit method are more robust, because when you invert the matrix you propagate information everywhere. I think the philosophy is that explicit methods are for hyperbolic, and implicit for parabolic. –  timur Sep 18 '11 at 14:55
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