Suppose we have two functions $f_1=f_1(t,x,v)$ and $f_2=f_2(t,x)$ and we do a finite volume scheme over control volumes $V_j\times X_i$ with respect to $(v,x)$. In time, one uses an implicit Euler scheme.
Suppose, we have already done the implicit Euler scheme in time and now we want to use a finite volume scheme with respect to $x$ and $v$.
Is it a valid step to do the following: $$ \begin{align*} \iint_{V_j\times X_i}f_2(t^{n+1},x)f_1(t^{n+1},x,v)\, dx\, dv&=\iint_{X_i\times V_j}f_2(t^{n+1},x)f_1(t^{n+1},x,v)\, dv\, dx\\ &=\int_{X_i}f_2(t^{n+1},x)\underbrace{\int_{V_j}f_1(t^{n+1},x,v)\, dv}_{\Delta v_jf_{1,j}^{n+1}(x)}\, dx\\ &=\Delta v_j\underbrace{\int_{X_i}f_2(t^{n+1},x)f_{1,j}^{n+1}(x)\, dx}_{\Delta x_i f_{2,i,j}^{n+1}f_{1,i,j}^{n+1}}\\ &=\Delta x_i\Delta v_j f_{1,i,j}^{n+1} f_{2,i,j}^{n+1} \end{align*} $$
In other words: I first swap the order of integration since $f_2$ is independent of $v$. Then I do a finite volume scheme for $f_1$ (inner integral), but only with respect to $v$ which gives me a function in $x$ which I then have as a factor in the outer integral for which I finally do a finite volume approximation.
Is this reasonable or valid? I have less experience with finite volume schemes.
