discretization of the 2nd wave equation
 

Hi,
Can someone pls tell me how the 2nd order wave equation is discretized using the finite volume method? Any references would be appreciated.
Thanks,

In order to use FV method, you have to start form a conservative form of the governing equation. That is:

- differential conservative form ... Div (Flux) = ...
- Integral form ........................ Int [S] ( n . Flux) dS = ...

I suggest the book of Leveque on FV for hyperbolic equations

I'll check the book. Tanx.

I know the general rules of FVM just confused with being 2nd order in time and space and would appreciate examples and references.

not all systems of hyperbolic equations can be solved with FVM ... for example, Euler equation in primitive variable [rho, u, p] have not a conservative form, you must go back to the original system in [rho, rho*u, rho*E]

So you're saying that the 2nd order wave equations cannot be solved using FVM?

Assuming the equation is in the form Phi_xx - Phi_yy=0,

you should define a divergence-like operator Div = (d/dx, d/dy) such that

Div F = 0

with F = ( u, v) where u = dPhi/dx, v = - dPhi/dy

and add an equation to close the problem

du/dy + dv/dt = 0

the equation is U_tt= c² LAP(U) where U_tt is the second order time derivative and LAP is the laplacian operator.
As Filippo said it for spatial derivative integration is straightforward because LAP(U) = DIV(Grad(U))

For the time derivative discretization just use a forward finite difference formulae for U_tt multiplied by the volume of the cell.

Thank you both. I think I more or less understand. So we need to make the spatial derivative to 1st order?

The way you have to integrate the laplacian using finite volume may be found in any text book. check for Poisson equation using finite volume.
For the second order time derivative,you have to find a forward expression based on finite difference and you will have to multiplied it by the volume of the cell that's all.;)

As leflix said, assuming your equation is Phi_tt = Div Grad (Phi), you integrate over a volume V of boundary S and apply the Gauss theorem to write

Int [V] Phi_tt dV = Int [S] (n . Grad (Phi) ) dS

This equation can be discretized at second order in time and space, for example using central derivatives. How course other issues as numerical stability, numerical oscillations, ecc have to be taken into account

Thank you bother very much.