Title: Two-level schemes for the advection equation,
Abstract: The advection equation is the basis for mathematical models of continuum
mechanics. In the approximate solution of nonstationary problems it is
necessary to inherit main properties of the conservatism and monotonicity of
the solution. In this paper, the advection equation is written in the symmetric
form, where the advection operator is the half-sum of advection operators in
conservative (divergent) and non-conservative (characteristic) forms. The
advection operator is skew-symmetric. Standard finite element approximations in
space are used. The standart explicit two-level scheme for the advection
equation is absolutly unstable. New conditionally stable regularized schemes
are constructed, on the basis of the general theory of stability
(well-posedness) of operator-difference schemes, the stability conditions of
the explicit Lax-Wendroff scheme are established. Unconditionally stable and
conservative schemes are implicit schemes of the second (Crank-Nicolson scheme)
and fourth order. The conditionally stable implicit Lax-Wendroff scheme is
constructed. The accuracy of the investigated explicit and implicit two-level
schemes for an approximate solution of the advection equation is illustrated by
the numerical results of a model two-dimensional problem.