New answers tagged finite-volume
0
That function looks pretty well behaved/smooth. So assuming your voxels are pretty small compared to the surface curvature, I think you could get away with a signed-distance-like approach, by testing the 8 corners against the surface. This will quickly eliminate the all-corners-inside (1) and all-corners-outside (0) cases.
Near the surface there will be ...
1
It's clear that the mass of $v$ is not conserved in general. Just take $u(x,t=0) = 0$ and choose $f$ so that $f(0)=0$. Then
$$\left. \frac{\partial v}{\partial t}\right|_{t=0} = -\frac{v}{\epsilon^2}.$$
2
Yes, mass conservation holds, because in fact you are solving some sort of a relativistic diffusion equation. Why:
$$\partial_{t} u + \partial_{x} v = 0$$
and:
$$\partial_{t} v + \frac{1}{\epsilon^{2}} \partial_{x} u = -\frac{1}{\epsilon^{2}} (v - f(u))$$
But from first equation:
$$\partial_{tt} u + \partial_{xt} v = 0$$
and from the second one:
$$\...
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