Linear schemes with several degrees of freedom for the 1D transport equation

We consider linear schemes with several degrees of freedom for the 1D 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 $q\geqslant p$ (for the discontinuous Galerkin method $p = k+1$ and $q = 2k+1$ where $k$ is the order of polynomials). We prove that this estimate holds if and only if there exists 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 $O(h^p)$. This fact leads to an algorithm establishing the optimal values $p$ and $q$ for a given scheme.