Suppose we use a finite volume method in the following sense:
Restrict the velocity domain to a bounded symmetricsegment $[−v^∗, v^∗]$ . We consider a mesh of this interval composed of $2L$ control volumesarranged symmetrically around $v = 0$ . We denote $v_{j+\frac{1}{2}}$ the $2L+1$ interface points with $j\in J=\{-L,\ldots,L\}$. In this sense,$$v_{-L+\frac{1}{2}}=-v^*,\quad v_{\frac{1}{2}}=0,\quad v_{j+\frac{1}{2}}=-v_{j-\frac{1}{2}}~\forall j=0,\ldots,L.$$For the sake of simplicity, we assume that the velocity mesh is uniform, namely that everycell $V_j=(v_{j-\frac{1}{2}},v_{j+\frac{1}{2}})$ has constant length $\Delta v$. Denoting $v_j$ the midpoint of cell $V_j$, we also have $v_j=-v_{-j+1}$ for all $j=1,\ldots,L$.
In space, we consider a uniform discretization of the torus $\mathbb{T}$ into $N$ cells$$X_i=(x_{i-\frac{1}{2}},x_{i+\frac{1}{2}}),\qquad i\in I:=\mathbb{Z} / N\mathbb{Z}$$of length $\Delta x$.
The control volumes of the phase space are then defined by$$K_{i,j}=X_i\times V_j\qquad (i,j)\in I\times J.$$Finally, we set $\Delta t>0$ the time step, and $t^n=n\Delta t$ for all $n\geq 0$.
My question is the following:
Suppose, in this setup, we have
$$\sum_{(i,j)\in I\times J}\frac{\Delta v v_j}{2}(f_{i+1,j}^n-f_{i-1,j}^n)\ln\frac{f_{i,j}^n}{c}\tag{*}$$for some positive constant $c>0$ and where $f_{i,j}^n$ is the approximation of the positive function $f=f(x,v,t)$ with respect to the lattice. Can we say something about the sign of this expression? I would like to have this expression non-negative. Maybe, if it is negative though, there is at least a way to keep it "arbitrary small".
Note that$$\sum_{(i,j)\in I\times J}\frac{\Delta v v_j}{2}(f_{i+1,j}^n-f_{i-1,j}^n)\frac{f_{i,j}^n}{c}=0\tag{**}$$due to the spatial periodic boundary conditions.
Is there some possibility to relate this to $(*)$?
My idea was that if$$f_{i,j}^{n}\approx f_{i\pm 1,j}^n$$(whatever this means precisely, maybe Taylor expansion at first order, i.e., something like $f_{i,j}^n=f_{i\pm 1,j}^n+(D_x f_j^n)_{i\pm 1}\Delta x$ for some discrete gradient $D_x$...)
then we can write $(*)$ as$$\begin{align*}&\sum_{(i,j)\in I\times J}\frac{\Delta v v_j}{2}(f_{i+1,j}^n-f_{i-1,j}^n)\ln\frac{f_{i,j}^n}{c}\\&\approx \sum_{(i,j)\in I\times J}\frac{\Delta v v_j}{2}(f_{i+1,j}^n-c)\ln\frac{f_{i+1,j}^n}{c}+\sum_{(i,j)\in I\times J}\frac{\Delta v v_j}{2}(c-f_{i-1,j}^n)\ln\frac{f_{i-1,j}^n}{c}\end{align*}$$
The first term on the right-hand side should be non-negative, while the second one is non-positive.
I have no idea whether this is of any help to answer my question.