# FVM - virtual node discretisation

I have come across the paper titled: A monolithic fluid structure interaction algorithm ...

For a 1D grid, at the boundaries the paper uses virtual nodes $x_{0}$ and $x_{N+1}$ (page 372) and for instance for velocity v at the boundaries ends up via extrapolation with (page 374)

$x_{0} = 2 x_{1}-x_{2}$

and

$x_{N+1} = 2 x_{N}-x_{N-1}$

I was wondering if someone could explain to me where the points N and 0 lie? On the faces or at nodes? Also what is the reason for using this extrapolation.

Thank you!

The finite volume method (FVM) uses cell averages to represent the numerical solution. For example, $x_i$ indicates the center of the $i$-th grid cell, and $\textbf{U}_i^n$ denotes the numerical solution inside grid cell $i$ at time $t^n$. The fluxes at cell boundaries are indicated by $i \pm 1/2$ subscripts, e.g., using your paper's notation, $\Phi_{i+1/2}$ is the numerical flux at the right boundary of the $i$-th cell (at grid coordinate $x_{i+1/2}$).
Your grid cell indices range from $1$ to $N$. In addition to these, the authors define two "ghost" cells with indices $0$ and $N+1$. These are not part of your "real" mesh configuration. Instead, their definition allows you to compute the fluxes at the "real" boundaries of your grid (i.e., $\Phi_{1/2}$ and $\Phi_{N+1/2}$). In 1D, you can think of cell no $0$ as being immediately to the "left" of grid cell $1$. Similarly, cell $N+1$ lies immediately to the "right" of cell $N$, like in the illustration given here.
• Thanks, just so that I am clear, essentially I need to add two extra nodes to my computational grid (two extra lines of codes for the boundary terms) and say at the left BC we have $\phi_{0} = 2 \Phi_{1} − \Phi{2}$. From that then for instance if we are averaging at the boundary, $\Phi_{1/2}=0.5 (\Phi_0 + \Phi_1$) which if I'm not mistaken gives the equation I mentioned before $\Phi_{1/2} = (3/2) \Phi_1 - (1/2) \Phi_2$ , but I guess that is only true if a linear interpolation is used at the BC's, and you're right it is different if another method is used at the BC ? Sep 7 '15 at 10:17
• @Hooman There is some confusion here. $\Psi_{i\pm 1/2}$ are the fluxes (defined only at the cell boundaries $x_{i \pm 1/2}$) and $U_i$ is the numerical solution. There is no such thing as $\Psi_{1}$, for example. The extrapolation formulas apply to $U_i$ (not to $\Psi$). From the $U_i$ you calculate the fluxes using, e.g., first order upwind splitting (equation 18 in your reference). Sep 7 '15 at 10:28
• I am still confused as in the reference (equation 34) it is setting $w_0(2) = \rho_0 U_0 = 0 - \rho_2 U_2$ so essentially it is setting velocity at i = 0 to 0 which does not make sense if i is a cell centre, as velocity is 0 at the wall (i.e. cell face). Please help! I am sure this is something simple that I don't get! Sep 7 '15 at 21:35
• @Hooman: Yes, the grid (coordinate) velocity variables $w_i$ are calculated at cell boundaries (grid coordinates $x_{i\pm1/2}$). Note that these are not the same as the contravariant velocities $U_i := u_i - w_i$. Sep 8 '15 at 7:48