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 conservation and monotonicity of the solution.

In this talk, the advection equation will be 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 standard explicit two-level scheme for the advection equation is absolutly unstable. New conditionally stable regularized schemes are constructed. Based on the general stability theory (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 orders.

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.