标题： 高阶通量分裂格式求解气体动力学欧拉方程 显示英文标题

标题： High-Order Flux Splitting Schemes for the Euler Equations of Gas Dynamics

摘要： 我们为一维和二维气体动力学的欧拉方程开发了高阶通量分裂格式。所提出的格式是现有的一阶通量分裂格式的高阶扩展，这些格式最初在 [E. F. Toro, M. E. Vázquez-Cendón, 计算与流体力学, 70 (2012), pp. 1--12] 中引入，其中气体动力学的欧拉方程被分解为两个子系统：对流系统和压力系统。本文首次在半离散框架内提出了TV分裂的高阶扩展。通过使用分段线性插值来重构未知变量的一侧点值，得到了二阶扩展。三阶和五阶格式则采用有限差分替代加权本质上非振荡（A-WENO）框架进行开发，该框架特别适用于处理多维问题，并且提供了构造高阶WENO格式更直接的方法。这些扩展显著提高了间断分辨率和数值解的精度，这一点通过一维和二维气体动力学欧拉方程的一系列数值实验得到了验证。 显示英文摘要

摘要： We develop high-order flux splitting schemes for the one- and two-dimensional Euler equations of gas dynamics. The proposed schemes are high-order extensions of the existing first-order flux splitting schemes introduced in [ E. F. Toro, M. E. V\'azquez-Cend\'on, Comput. \& Fluids, 70 (2012), pp. 1--12], where the Euler equations of gas dynamics are split into two subsystems: the advection and pressure systems. In this paper, we formulate the TV splitting within the semi-discrete framework to extend it to higher orders of accuracy for the first time. The second-order extension is obtained by using piecewise linear interpolant to reconstruct the one-sided point values of the unknowns. The third- and fifth-order schemes are developed using the finite-difference alternative weighted essentially non-oscillatory (A-WENO) framework, which is particularly effective in handling multidimensional problems and provides a more straightforward approach to constructing higher-order WENO schemes. These extensions significantly improve the resolution of discontinuities and the accuracy of numerical solutions, as demonstrated by a series of numerical experiments of both the one- and two-dimensional Euler equations of gas dynamics.