浅水波方程组的熵稳定有限体积格式

Entropy Stable Finite Volume Scheme for Shallow Water Equations

摘要: 本文针对非平底地形上的浅水波方程组，提出了一种高精度熵稳定有限体积格式。首先，我们构造了一个具有二阶精度的well-balanced的半离散熵守恒格式，该格式满足给定熵对的熵恒等式，并精确地保持静水稳态。本文的关键点是使通量梯度和源项的离散化相互匹配；其次，二阶熵守恒格式的可负担熵守恒通量对于最终的高阶格式也是至关重要的。然后，以二阶熵守恒格式为基本模块，实现了高阶well-balanced的半离散熵守恒格式。第三，通过在现有的熵守恒格式中添加适当的耗散项，提出了高阶精度的well-balanced的半离散熵稳定格式，该耗散项基于标度熵变量的加权本质非振荡重构，以克服熵守恒的数值振荡。最后，使用Runge-Kutta方法对半离散格式进行时间积分，以实现最终格式。用大量的数值结果说明所提出的格式满足离散熵不等式，具有良好的平衡性，保持了光滑解的真正高阶精度，并且能够很好地捕获稳态下的小扰动。

Abstract: This article develops a high-order accurate entropy stable finite volume scheme for the shallow wa-ter equations over non-flat bottom topography. Firstly, we construct a second-order accurate well-balanced semi-discrete entropy conservative scheme, which satisfies the entropy identity for the given entropy pair and maintains the still water steady states exactly. The key idea is to make both discretizations for the flux gradient and the source term match each other. Secondly, afforda-ble entropy conservative fluxes of the second-order entropy conservative schemes are also critical for the ultimate high-order scheme. Then, high-order well-balanced semi-discrete entropy con-servative scheme is achieved by taking the second-order entropy conservative schemes as a build-ing block. Thirdly, the high-order accurate well-balanced semi-discrete entropy stable schemes are proposed by adding a suitable dissipation term to the existing entropy conservative scheme based on the weighted essentially non-oscillatory reconstruction of the scaled entropy variables to sup-press the numerical oscillations of the entropy conservative scheme. Finally, the semi-discrete scheme is integrated in time using the Runge-Kutta approach to achieve the eventual scheme. Ex-tensive numerical results strongly indicate that the proposed scheme satisfies the discrete entropy inequality, is well-balanced, keeps the genuine high-order accuracy for smooth solutions, and can well capture small perturbations according to the steady state.