Wonky solutions for mixed parabolic-hyperbolic PDEs

I'm solving the equations:

I'm using the finite volume method with a constant spacing on dh, so it seems to be a simple solution. The boundary conditions are functions of \nu, and I compute the boundary terms by using a simple polynomial. I conpute the density which is 1/\nu, but I get funky boundary values that I don' understand. can anyone explain what is wrong?

[Torsten]

Isn't it easier to solve the equivalent system

dv/dt = vdot
dvdot/dt = d^2/dh^2 (zeta(v) * vdot)

?

I get this from

d^2v/dt^2 = d/dt (du/dh) = d/dh (du/dt) = d^2/dh^2 (zeta(v)*du/dh) = d^2/dh^2 (zeta(v)*dv/dt)
         (1)                           (2)                        (1)         

where (1) and (2) refer to your first and second equation from above.

[Mat Hunt]

How will that help?

The idea I used was to average over a cell, and then approximate the average by the value at the centre of the cell so I get the integrated equation:

I Then use ODE15 to solve these equations.

[Torsten]

How will that help?

In my opinion, it helps as the first PDE becomes an ODE and to see what boundary conditions are necessary to fix a solution, namely prescribing v and vdot over [0,L] at t=0 and prescribing values that include vdot or dvdot/dh at h = 0 and h = L.

For me, your code is hard to read. Which boundary conditions do you try to impose in the formulation given ? I think this the hardest part: to know which boundary conditions are necessary and sufficient to solve this type of PDE system.