Question about FDM and supersonic flows
I have a naive question about Finite Differences Method in the case of supersonic flows.
Correct me if I'm mistaken but when deriving the differential form of the NS equations from the integral form, the use of the divergence theorem implies that the velocity field need to be continuous and differentiable. Therefore, it seems to me that FDM, that come directly from the differential form of the NS equation, are irrelevant to simulate compressible flows with shocks that are, by nature, discontinuous. Yet there are many studies based on FDM for compressible flows.
Am I missing something here ? Is there a numerical artifact I am not aware of that can compensate this mathematical irrelevance ? Is it really mathematically irrelevant ?
Thanks for your help
your doubts are right and I can address some issues, hoping they can be useful. Sorry if I will be lenghty in the response...
As you correctly said, the integral (weak) form of conservation law is the only one for which discontinuous solution can be managed. The differential form has no meaning across a discontinuity.
However this is what one can say for the continuous form of the equations and for the Euler equation for which a singularity can appear also from regular initial condition.
Before discussing the discrete form, let me first use the 1D Burgers equation as example:
Differential (strong) form
du/dt + d/dx(u^2/2)=0 in a periodic domain
u(x,0) = u0(x)= C*sin(x) initial condition
the solution is regular until the time tb = -1/min(u0'(x)), then, you must consider that weak form. The local averaged form can be demonstrated to be a weak form for a particular choice of the test function.
Then, if you integrate the differential Burgers equation over a local volume of measure h centred around x, use the Gauss theorem you get
Integral (weak) form
d u_bar/dt + [u^2(x+h/2,t) - u^2(x-h/2,t)]/2h=0
with u_bar = (1/h) int [x-h/2, x+h/2] u(x',t) dx'.
The issue of having the closing relation u=u[u_bar] is called flux reconstruction from averaged cells. The most simple reconstruction can be to assume u = u_bar, that is accurate to O(h^2) for centred cell.
Now the problem is the "confusion" one can find in several book that use the Finite Difference (FD) or Finite Volume (FV) approximation.
FD discretizations are only for the differential form of the equations and FV for the integral ones. But...
1) Traditionally second order central discretization were used, then if you see the discrete equation:
dU/dt (i,n) + [U^2(i+1,n) - U^2(i-1,n)]/2h =0
it looks like a FD discretization... but if you start from the integral formulation to develop a FV method and use a linear reconstruction for the fluxes you will get the same discrete equation! Thet are a consistent discretization to the integral form of the conservation laws. For discretizations at higher order, FD and FV form can no longer be confused.
2) Sometime the use of FD or FV appears misleading in reading some textbooks. To clarify the point, you can write a differential form for the pointwise field u(x,t) but you can also write a differential form governing the averaged (weak) field u_bar(x,t). Owing to the averaging operator, the field is regularized. This is practically the same thing that leads to write the LES equations: differential equations for a filtered field.
In conclusion, if you read FD in cases where inviscid flows with singularity are treated, you must not think of the discretization of the strong form of the equation but is related to the weak one.
My opinion is that is better to refer to FV formulation coming from the integral form of the equations but I know that the literature is not univocal about this point.
You can find some reading in the book of LeVeque on FV methods as well as in the book of Peric and Ferziger
Is this what you're trying to say? http://hdl.handle.net/2060/19880020687 :) From ntrs.nasa.gov.
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