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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!

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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.

The solution approximation inside the ghost cells is computed using standard extrapolation formulas.

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  • $\begingroup$ 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 ? $\endgroup$ – Hooman Sep 7 '15 at 10:17
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    $\begingroup$ @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). $\endgroup$ – GoHokies Sep 7 '15 at 10:28
  • $\begingroup$ 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! $\endgroup$ – Hooman Sep 7 '15 at 21:35
  • $\begingroup$ @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 $U_i := u_i - w_i$. $\endgroup$ – GoHokies Sep 8 '15 at 7:48
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    $\begingroup$ No I haven't as things still don't add up ... Although I think that the extrapolation is a general equation that can be used between any equally spaced points, whether they are at cell faces or nodes. I would appreciate it if u updated the answer, hopefully it would make things clear. (I deleted some of my short replies). $\endgroup$ – Hooman Sep 8 '15 at 13:47

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