基于广义Hermite展开的全局双曲矩方程组  

Globally hyperbolic moment system by generalized Hermite expansion

作　　者：樊玉伟[1] 李若[1] 

机构地区：[1]北京大学数学科学学院,北京100871

出　　处：《中国科学：数学》2015年第10期1635-1676,共42页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11325102和91330205)资助项目

摘　　要：在最近的研究中,Cai等人(2014)使用本征热通量代替通常使用的沿坐标轴方向的热通量作为变量获得了一个改进的13矩方程组.该方程组比使用坐标轴方向的热通量获得的Grad 13矩方程组具有更好的性质,例如,局部平衡态是双曲区域的内点.该改进的13矩方程组是通过对分布函数使用广义的各向异性Hermite展开获得,其中该各向异性展开是指,使用完全的温度张量代替局部平衡态的温度.本文将该方法推广到高阶广义Hermite展开从而获得任意阶的新的矩方程组,并类似Cai等人(2014)的方法提出了一个正则化方案,使得所得方程组全局双曲,从而保证所得方程组的局部适定性.此外,本文还深入研究该方程组的系数矩阵的特征结构,并剖析了方程组的所有特征波.该矩系统提供了一套系统的可看成是Euler方程组的推广的动力学模型,并可通过提高展开阶来逼近Boltzmann方程本身.In a recent paper,Cai et al.（2014） revealed that a modified 13-moment system taking intrinsic heat fluxes as variables,instead of the heat fluxes along the coordinate vectors which is adopted in the classical Grad13-moment system,attains some additional advantages than the classical Grad 13-moment system,particularly including that the equilibrium is turned to be the interior point of its hyperbolicity region. The modified 13-moment system was actually derived from the generalized Hermite expansion of the distribution function,where the anisotropy of Hermite expansion is specified by the full temperature tensor. We extend the method therein in this paper to high order of generalized Hermite expansion to derive arbitrary order moment systems,and propose a globally hyperbolic regularization to achieve locally well-posedness similar to the method of Cai et al.（2014）.Furthermore,the structure of the eigen-system of the coefficient matrix and all characteristic waves are fully clarified. The obtained systems provide a systematic class of hydrodynamic models as the refined version of Euler equations,which is gradually approaching the Boltzmann equation with increasing order of the expansion.

关 键 词：动力学模型 矩方法 全局双曲性 正则化 NRxx 

分 类 号：O175[理学—数学]