Can anyone explain how, through semi-discretization, you can transform a PDE into an ODE?

Also, a brief explanation of finite volume discretization would be most helpful.

Thanks,

Nick

If you have a PDE where time and x are the independent variables, and you discretize only in x then you end up with a set of ODE's in time at each spatial point xi.

Finite volume discretization involves breaking up the domain into a set of non-overlapping subdomains that completely cover the original domain and writing the equations of fluid motion for each of those subdomains. Since the subdomains are usually geometrically simple (tets, prisms, etc.) the flux integrals reduce to the sum of the face areas times the assumed fluxes, which are computed by interpolating nodal or cell-center values. By keeping the time derivative of the volume integral undiscretized, you end up with a set of ODE's, one per subvolume.

Is this method the so called 'Method of Lines'? What is it's validity? How closely does the ODE represent the PDE? How can you analyse this? Thanks.

It appears that it can be related to the method of lines, as per approaches such as shown here:

www.polymath-software.com/papers/cachen2.pdf

With regard to validity, I would guess that it is as valid as the discretization approximation allows it to be. I'm not sure what you mean by validity.

Yes, it is often referred to as the method of lines. The semi-discrete ODE is a "valid" representation of the PDE if it is consistent with the analytical equation, i.e. if the spatial discretization error approaches zero as the grid resolution is increased. "Validity" in this sense does however not guarantee stability. The discretization also has to be stable in order to achieve convergence.