Three-Level Schemes for the Advection Equation

Abstract

The advection equation, which is central to mathematical models in continuum mechanics, can be written in the symmetric form in which the advection operator is the half-sum of advection operators in the conservative (divergence) and nonconservative (characteristic) forms. In this case, the advection operator is skew-symmetric for any velocity vector. This fundamental property is preserved when using standard finite element spatial approximations in space. Various versions of two-level schemes for the advection equation have been studied earlier. In the present paper, unconditionally stable implicit three-level schemes of the second order of accuracy are considered for the advection equation. We also construct a class of schemes of the fourth order of accuracy, which deserves special attention.