标题： 差分流体力学格式在冲击波存在下的收敛性特性 显示英文标题

标题： Convergence properties of finite-difference hydrodynamics schemes in the presence of shocks

摘要： 我们研究在$\Delta x \!\rightarrow\! 0$极限下的渐近收敛性，作为一种工具，用于确定涉及激波的数值计算是否准确。 我们使用一维算子分裂有限差分方案进行流体力学计算，并采用冯·诺伊曼人工粘性。 一种内能方案收敛到明显错误的解。 我们将这种失败归因于极限解中存在不连续性。 我们对拉克斯-温德罗夫定理的扩展保证了某些保守的、算子分裂的方案收敛到正确的连续介质解。 对于应用于单个激波形成的总能量方案，柯西误差的收敛速度缓慢地接近预期速率。 我们将这种缓慢性与由于变化的线性人工粘性长度导致的扩散变化对小振幅波的影响联系起来。 在附录中，我们讨论了激波过渡区域与粘性长度的标度关系，并展示了试图进行外推时遇到的几种困难。 显示英文摘要

摘要： We investigate asymptotic convergence in the~$\Delta x \!\rightarrow\! 0$ limit as a tool for determining whether numerical computations involving shocks are accurate. We use one-dimensional operator-split finite-difference schemes for hydrodynamics with a von Neumann artificial viscosity. An internal-energy scheme converges to demonstrably wrong solutions. We associate this failure with the presence of discontinuities in the limiting solution. Our extension of the Lax-Wendroff theorem guarantees that certain conservative, operator-split schemes converge to the correct continuum solution. For such a total-energy scheme applied to the formation of a single shock, convergence of a Cauchy error approaches the expected rate slowly. We relate this slowness to the effect of varying diffusion, due to varying linear artificial-viscous length, on small-amplitude waves. In an appendix we discuss the scaling of shock-transition regions with viscous lengths, and exhibit several difficulties for attempts to make extrapolations.