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Two-level schemes for the advection equation

机译：平程方程的两级方案

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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 standard explicit two-level scheme for the advection equation is absolutely 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. (c) 2018 Elsevier Inc. All rights reserved.
机译：对流方程是连续介质力学数学模型的基础。在非平稳问题的近似解中，有必要继承解的保守性和单调性的主要性质。本文将对流方程写成对称形式，其中对流算子是保守（发散）和非保守（特征）形式的对流算子的半和。平流算子是斜对称的。使用空间中的标准有限元近似。平流方程的标准显式两层格式是绝对不稳定的。构造了新的条件稳定正则化格式，基于算子差分格式稳定性（适定性）的一般理论，建立了显式Lax-Wendroff格式的稳定性条件。无条件稳定和保守格式是二阶（Crank-Nicolson格式）和四阶的隐式格式。构造了条件稳定的隐式Lax-Wendroff格式。模型二维问题的数值结果表明，所研究的对流方程近似解的显式和隐式两层格式的精度。（c） 2018爱思唯尔公司版权所有。

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