Three-Level Schemes for the Advection Equation
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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.
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