According to this paper the following finite difference approximation is third-order accurate:
$$\frac{d\rho_j}{dx}\approx\frac{2-\eta}{3}\frac{\rho_{j+1/2}-\rho_{j-1/2}}{\Delta x}+\frac{1+\eta}{3}\frac{\rho_{j-1/2}-\rho_{j-3/2}}{\Delta x}\text{, with }\eta=\frac{u\Delta t}{\Delta x}.$$
But I don't see how this is third-order accurate. If you Taylor expand $\rho_j$ you get:
$$\frac{\rho_{j+1/2}-\rho_{j-1/2}}{\Delta x}=\frac{d\rho_i}{dx}+\frac{\Delta x^2}{24}\frac{d^3\rho_j}{dx^3}+O(\Delta x^3),$$
$$\frac{\rho_{j-1/2}-\rho_{j-3/2}}{\Delta x}=\frac{d\rho_i}{dx}-\Delta x\frac{d^2\rho_j}{dx^2}+\frac{13\Delta x^2}{24}\frac{d^3\rho_j}{dx^3}+O(\Delta x^3).$$
Therefore, at best this scheme is first-order accurate? For reference see page 6 of the paper that is linked. Or this image here: