Linear schemes with several degrees of freedom for the transport equation and the long-time simulation accuracy

Abstract

Abstract We consider linear schemes with several degrees of freedom for the transport equation on uniform meshes. For these schemes, the solution error is $O(h^p + th^q)$, where $p$ is equal to or greater by one than the order of the truncation error 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 $O(h^p)$. In a one-dimensional case, this mapping can be found in the class of local mappings. In more dimensions, the existence of a local mapping with such properties is guaranteed only under additional assumptions.