一类基于非等距单元平均重构的高效差分格式

A class of high resolution difference schemes based on non-uniformly cell averaged-solution reconstruction

构造了一维非线性双曲型守恒律的一类基于非等距单元平均值的点值重构的高精度高分辨率守恒型差分格式。其构造思想是:首先,将计算区间划分为若干个互不重叠的小区间,再根据格式精度的要求利用Gauss-Lobatto点和Gauss-Chebyshev点划分小区间,通过各非等距细小区间上的单元平均值,重构各细小区间交界面上的点值,并加以校正;其次,利用近似Riemann解计算细小区间交界面上的数值通量,并结合高阶Runge-Kutta TVD时间离散方法,得到了高精度的全离散方法。证明了该格式的无振荡特性。然后,将格式推广到一维双曲型守恒方程组情形。最后,给出了几个标准数值算例,验证了格式的高效性。

Based on non - uniformly cell averaged - solution reconstruction, a class of high - order accuracy and high resolution conservative difference schemes is obtained for one - dimensional nonlinear hyperbolic conservation laws in this paper. Its idea is as follows. First, the computational interval is divided into pieces of nonoverlapping sub - intervals, and then each sub - interval is further subdivided into small - intervals by using Gauss - Lobatto and Gauss - Chebyshev partitions according to the required accuracy. Cell averaged - solutions from these small - intervals are used to reconstruct solutions at small - interval boundaries. Furthermore the correction is introduced. Second, the approximate Riemann solver is used to compute numerical fluxes at small - intervals boundaries, and a high - order accurate full discretization method is obtained by applying high - order Runge - Kutta TVD time discretization. Moreover, the non - oscillatory property of the scheme is proved. The extension to systems is implemented. Finally, several typical numerical experients are given. The numerical results verify high accuracy and high resolution of the resulting schemes.