全隐式TVD格式求解二维Euler方程的对角化算法

本文采用有限体积法,Chakravarthy提出的基于近似Riemann解的全隐式TVD差分格式求解二维非定常Euler方程。采用牛顿法对半离散化后的全隐式差分方程进行线性化,然后对线性化后的方程采用近似因式分解法处理,得到了两组三对角块矩阵方程组。在求解该方程组时,采用了矩阵对角化思想使原来的4×4三对角块矩阵方程组转化为4组分离的三对角代数方程组,因而大大节省了求解方程组所耗费的计算时间,提高了计算效率。通过对翼型的跨声速绕流问题及圆柱超声速流动问题的计算,证实了本文算法不但简便可靠,而且还具有较强的通用性。采用TVD格式捕捉到的激波分辨率高,上、下游无任何波动。

Calculation of Two-Dimensional Euler Equations Using an Implicit TVD Scheme by Matrix Diagonalization

In this paper, two-dimensional Euler equations are solved by finite volume method using a TVD (Total Variation Diminishing) scheme proposed by Chakravarthy which is based on Roe's Approximate Riemann Solver.A Newton procedure is used in the linearization of the fully implicit equations. After linearization an approximate factorization approach is used,The induced block matrix equations are transformed into separated algebraic equations by matrix diagonalization. Thus the computer time needed for solving the equations is much reduced.The computaional results of transonic flow about an airfoil and supersonic flow about a cylinder are given in this paper. The numerical results show that the method presented in this paper is indeed time-saving and in good agreement with the known results.The results also show the good shock capturing ability and high shock resolutions of TVD scheme.