非线性双曲型守恒律的高精度MmB差分格式

构造了一维非线性双曲型守恒律方程的一个高精度、高分辨率的广义G odunov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相交的小区间,再根据精度要求等分小区间,通过各细小区间上的单元平均状态变量,重构各等分小区间交界面上的状态变量,并加以校正;其次,利用近似R iem ann解算子求解细小区间交界面上的数值通量,并结合高阶R unge-K u tta TVD方法进行时间离散,得到了高精度的全离散方法。证明了该格式的Mm B特性。然后,将格式推广到一、二维双曲型守恒方程组情形。最后给出了一、二维Eu ler方程组的几个典型的数值算例,验证了格式的高效性。

A high order accurate MmB difference scheme for nonlinear hyperbolic conservation laws

In this paper,a high-order accuracy,high resolution,generalized Godunov-type difference scheme is presented for 1D/2D nonlinear hyperbolic conservation laws.Firstly,the computational interval is divided into pieces of non-overlapping sub-intervals,and then each sub-interval is further subdivided into equal small-intervals according to required accuracy.Cell averaged-solutions from these small-intervals are used to reconstruct a high order polynomial approximation in small-interval boundaries. Furthermore the correction is introduced to prevent oscillations near discontinuities from the high-order approximation.Secondly,the approximate Riemann solver is used to compute numerical fluxs at small-intervals boundaries,and a high-order fully discretization method is obtained by applying high-order RungeKutta TVD time discretization.Moreover,we prove the MmB property of the scheme under a certain CFL condition,and extend to 1D/2D system of hyperbolic conservation laws.It does not necessitate the conventional field-by-field characteristic decomposition.Finally,several typical numerical experiments are given. The numerical results verify high resolution of the method.