带重力源项Euler方程的保平衡型加权紧致有限差分格式  

WELL-BALLANCED WEIGHTED NONLINEAR COMPACT SCHEME FOR EULER EQUATIONS WITH GRIVITATIONAL SOURCE

作　　者：唐玲艳[1] 刘涛 王志远 Tang Lingyan;Liu Tao;Wang Zhiyuan(Department of Mathematics,College of Sciences,National University of Defence Technology,Changsha 410073,China)

机构地区：[1]NUDT,国防科技大学理学院数学系,长沙410073

出　　处：《计算数学》2025年第1期135-148,共14页Mathematica Numerica Sinica

基　　金：湖南省自然科学基金面上项目(2024JJ5398)资助。

摘　　要：针对广义坐标系上带重力源项的Euler方程,提出一种基于加权紧致非线性差分方法的高阶保平衡型有限差分格式.基本思想是,利用稳态解信息对重力源项进行重构,使之在平衡状态下与流通量中的压力梯度形成对应关系;采用具有尺度不变性的非线性插值计算半节点处的守恒量,确保平衡状态下守恒量的重构值与稳态解的重构值精确相等;采用相同的中心差分格式计算通量导数和网格导数,保证数值格式在曲网格上满足几何守恒律.理论推导和数值试验证明了格式的保平衡性、保几何守恒律性,以及在曲网格能够获得高阶精度和对稳态解附近的小扰动实现精细捕捉.A new high-order well-balanced finite difference scheme based on weighted compact non-linear scheme(WCNS)is proposed for the Euler equation with gravitational source on the generalized coordinate system.The basic idea is to reconstruct the gravitational source term using steady-state solution,so that it can correspond to the pressure gradient at the left-hand-side of the equations in an equilibrium state.To ensure that the reconstructed value of the conserved variables is exactly equal to the reconstructed value of the steady-state solution in an equilibrium state,a nonlinear interpolation with scale invariance property is used in the reconstruction procedure.Since the same central difference scheme can be used for both flux derivatives and grid derivatives,the proposed scheme satisfy geometric conservation laws on curvilinear grids.By theoretical analysis and experimental results,it is indicated that the proposed WCNS scheme can preserve the general steady state which in-clude both ispthermal and polytropic equilibria,and geometric conservation laws.Moreover,it can achieve fifth-order accuracy and capture exactly small perturbations near steady-state solutions on curvilinear grids.

关 键 词：EULER方程 保平衡 加权紧致非线性格式 重力场 

分 类 号：O241.3[理学—计算数学]