Problem. Consider the one-dimensional adjoint Euler equations for $(x,t) \in \Omega \times [0,T]$ with $\Omega \subset \mathbb{R}$ and $T > 0$ $$ \varphi_t + \Big(\frac{\mathrm{d}F}{\mathrm{d} U}(x)\Big)^{\top}\ \varphi_x = 0 \quad \text{in} \quad \Omega \times [0,T] \tag{$*$} \label{eq:adj}$$ where $F$ is the flux vector of the 1D Euler equations, $U$ the conservative variable vector and $\varphi_x$ the derivative of the adjoint vector with respect to $x$.
In particular, \eqref{eq:adj} is the adjoint equation derived from the conservative form of Euler equations.
Question I. Now, what scheme, having a cell-centered finite volume (FV) discretization and artificial dissipation (similar to the JST scheme), would be appropriate to numerically solve this equation (considering that it is cast in a nonconservative form)?
Remark. If I am not mistaken, the above homogeneous and nonconservative form can be recasted to a nonhomogeneous and conservative form using the product rule of the derivative $$ \varphi_t + \Big[\Big(\frac{\mathrm{d}F}{\mathrm{d} U}\Big)^{\top}(x)\ \varphi \Big]_x = \Big[\Big(\frac{\mathrm{d}F}{\mathrm{d} U}(x)\Big)^{\top}\Big]_x\ \varphi$$
Question II. Is the above method correct? Also, considering that artificial dissipation is used, is there any faster and simpler way to discretize the system \eqref{eq:adj} avoiding numerical pitfalls?
Remark. The coefficients $\frac{\mathrm{d}F}{\mathrm{d} U}(x)$ are frozen in time $t$ and they depend only on $x$.