Linear schemes with several degrees of freedom for the multidimensional transport equation

We consider linear schemes with several degrees of freedom for the transport equation. The solution error possesses the estimate $O(h^p + th^q)$ where $p$ is equal to or greater by one than the truncation error order and $q\geqslant p$. We prove the existence of a mapping of smooth functions on the mesh space providing the $q$-th order of the truncation error and deviating from the standard mapping ($L_2$-projection for example) by the order $h^p$. In contrast with 1D case local mapping with such properties generally does not exist. We prove sufficient existence conditions.