求解静水稳态问题的保平衡高阶熵稳定格式  

Well⁃Balanced High⁃Order Entropy Stable Scheme for the Lake⁃at⁃Rest Steady⁃State Problem

作　　者：陈梦莹 郑素佩[1] 郭依琳 刘佳豪 高普阳 CHEN Mengying;ZHENG Supei;GUO Yilin;LIU Jiahao;GAO Puyang(College of Science,Chang'an University,Xi'an 710064,Shaanxi,China)

机构地区：[1]长安大学理学院,陕西西安710064

出　　处：《力学季刊》2024年第4期1018-1031,共14页Chinese Quarterly of Mechanics

基　　金：国家自然科学基金(11971075);陕西省自然科学基础研究计划资助项目(2024JC-ZDXM-23)。

摘　　要：具有广泛实际应用背景的静水稳态问题可由带源项的浅水波方程描述,数值计算中该类方程对求解格式的保平衡性要求严格.本文构造了基于低耗散TENO(Targeted Essentially Non-Oscillatory)重构的保平衡高阶熵稳定格式,证明了格式的保平衡性.将具有低数值耗散的TENO重构引入到TeCNO框架中,在已有熵守恒通量中加入重构后的数值耗散项,得到高阶熵稳定数值通量;源项部分采用与数值通量相对应的离散形式,以达到数值上平衡通量函数与源项的目的,通过若干数值算例验证算法的高分辨率与保平衡性.数值结果表明基于TENO重构的高阶熵稳定格式具有保平衡、低耗散和高分辨率的优点,能够很好地求解稳态解的小扰动问题.The lake-at-rest steady-state problem,which finds extensive practical applications,can be represented by shallow water equations with source terms.These equations necessitate a well-balanced approach in numerical calculations.This paper presents a well-balanced,high-order entropy stable scheme based on Targeted Essentially Non-Oscillatory(TENO)reconstruction for its low numerical dissipation.The well-balanced property of this scheme is subsequently established.The TENO reconstruction with low numerical dissipation is introduced into the TeCNO framework,appending the reconstructed numerical dissipation term to the existing entropy conservation flux to obtain high-order entropy stable numerical flux.For the source term,a discrete form corresponding to the numerical flux is adopted to achieve a numerical balance between the flux function and the source term.The high resolution and well-balanced property of the algorithm are demonstrated through multiple numerical examples.The numerical results show that the high-order entropy stable scheme,based on TENO reconstruction,offers the advantages of well-balanced property,low dissipation and high resolution.Furthermore,it effectively addresses small perturbation problem in steady-state solution.

关 键 词：静水稳态 保平衡性 熵稳定格式 TENO重构 浅水波方程 

分 类 号：O241.82[理学—计算数学]