Lax Wendroff Euler 2D in physical domain
 

Hello,

I discovered this form recently and have some question about the 2D FVM. Forgive me if this thread already exists but I don't know the correct keyword to look for.

I'm currently writing a solver for the 2D Euler equations in curvilinear coordinates and the later aim is to compute the flow around a turbine blade. The whole solver is supposed to work with Cartesian coordinates.

I want to use the Lax-Wendroff method and here is the catch since I want to write it directly in the physical domain. Unfortunately, I've got some problems with the flux computation and I'm some kind of confused how to formulate it properly.

Starting from a structured vertex centered grid (red points) I introduce a control (thick black line) volume by averaging with the neighbor points. The edges of this control volume are prescribed by the green points which come from the averaging of its surrounding 4 red neighbors.

By applying the divergence theorem I can say that the local temporal change of my conserved variable is equal to the sum of all fluxes through the CV's surface or as equation:

u_t = - (F_x + G_y) => u_t*dV = - sum[f*S*nx +g*S*ny]

, where S is the cell surface and nx and ny the normal vectors. Ok no problem so far but now comes the problem keeping me busy for almost one week:

Following Lax-Wendroff I add artificial viscosity in form of u_tt and express the second derivative in time by saying

u_tt = [- (F_x + G_y) ]_t =[- (A*u_x + B*u_y) ]_t = [- ([A*u_t]_x + [B*u_t]_y ]

Here I stuck because I haven't figured out yet how to do that for

- sum[f*S*nx +g*S*ny]

or vice versa how to formulate u_y and u_x for this CV?

I know my question is more a mathematical one but if you could recommend me any papers or have some good hints, I would really appreciate this.

Best regards,

Christian

You must use the Lax-Wendroff method for conservative laws. Give a look to the book Morton & Meyers, Numerical Solution of Partial Differential Equations, subsection 4.6 is what you are looking for.
Also the book of Leveque can be useful

Thank you for your post. Yes, I just looked it up but unfortunately it is only for 1D.

I don't know but might it be wiser to switch to metrics here? So I can apply all the scheme without worrying about the orientation...

you can use the FV flux reconstruction method showed in the book of Peric and Ferziger.
Furthermore, I worked on LW-like method on unstructured grids while using the integral formulation, you can see details in Int. J. Numer. Meth. Fluids 2002; 38:1133–1157 (DOI: 10.1002/
d.278). Specifically, see how the (16) to (18) are obtained and how the operators act on the fluxes