一类高精度TVD差分格式及其应用

High Order Accurate TVD Difference Scheme with Applications

构造了一维非线性双曲型守恒律的一个新的高精度、高分辨率的守恒型TVD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,重构各细小区间交界面上的状态变量,并加以校正;其次,利用近似R iemann解计算细小区间交界面上的数值通量,并结合高阶Runge-Kutta 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TVD特性。该格式适合于使用分量形式计算而无须进行局部特征分解。通过计算几个典型的问题,验证了格式具有高精度、高分辨率且计算简单的优点。

A class of conservative TVD (Total Variation Diminishing) difference schemes with high order accuracy and resolution, is presented for 1D nonlinear hyperbolic conservation laws. The computational interval is divided into pieces of nonoverlapping sub-intervals, and then each is further subdivided into identically small intervals according to the required accuracy. Cell averaged state variables from these small intervals are used to reconstruct a high order polynomial approximation in the small interval boundaries. Furthermore the correction is introduced to prevent oscillations near discontinuities from the high-order approximation. The approximate Riemann solver is used to compute numerical fluxs on small interval boundaries, and a high-order fully discretization method is obtained by applying high-order Runge-Kutta TVD time discretization. The new scheme enables to accelerate the computation with a higher accuracy and resolution.