In this paper we present a formally fourth-order accurate hybrid-variable method for the Euler equations in the context of method of lines. The hybrid-variable (HV) method seeks numerical approximations to both cell-averages and nodal solutions and evolves them in time simultaneously; and it is proved in previous work that these methods are inherent superconvergent. Taking advantage of the superconvergence, the method is built on a third-order discrete differential operator, which approximates the first spatial derivative at each grid point, only using the information in the two neighboring cells. Stability and accuracy analyses are conducted in the one-dimensional case for the linear advection equation; whereas extension to nonlinear systems including the Euler equations is achieved using characteristic decomposition and the incorporation of a residual-consistent viscosity to capture strong discontinuities. Extensive numerical tests are presented to assess the numerical performance of the method for both 1D and 2D problems.

翻译：本文在线法框架下提出了一种形式上四阶精度的混合变量方法，用于求解欧拉方程。混合变量方法同时求解单元平均值和节点解的数值近似，并将其随时间同步推进；先前研究已证明该方法具有固有超收敛性。利用这一超收敛特性，该方法基于三阶离散微分算子构建，仅需相邻两个单元的信息即可逼近每个网格点上的一阶空间导数。针对一维线性对流方程开展了稳定性与精度分析；通过特征分解及引入残差一致粘性捕捉强间断，实现了向包含欧拉方程的非线性系统的扩展。最后通过大量数值算例评估了该方法在一维和二维问题中的性能表现。