二维Euler方程无网格算法的精度分析

Accuracy analysis of gridless method for 2D Euler equations

在几种典型的点云布点方案中,分析了文献[3]中的无网格算法的计算精度。分析指出影响无网格算法计算精度的主要因素是点云中点的分布是否平衡;增加点云中节点的数目或者采用高阶曲面拟合都不能提高精度;这里的理论分析为无网格算法的布点提供了重要参考。

The potential for using a gridless method based on free of mesh concept for the Euler equations is investigated. The Euler equations are solved directly by using local least-squares curves-fits in each cloud of points, and then computing the curves-fits equations to approximate the spatial derivatives. This paper focuses on the accuracy analysis of this method. A few kinds of the point distribution within the cloud of points are specified to analyze the accuracy of the spatial derivatives approximation. This paper points out that the balance of points distribution within the cloud of points plays a leading role on the accuracy. A gridless method can achieve second-order accuracy on the balanced points distributions, but the accuracy is only first-order on the unbalanced point distributions. Increasing the number of points within the cloud of point or using the high-order curves-fits cannot improve the accuracy. The numerical experiment was conducted to verify the above conclusion. We use points filled by the two-dimensional structure grid generator to form two kinds of point cloud, one is balanced the other is unbalanced. The result indicates that the second-order accuracy is achieved on balanced point distribution since the pressure distribution is well agreement with that of second-order finite-volume method. But only the fist-order accuracy on the unbalanced points distributions was achieved, the pressure distribution also shows quite difference between second-order finite-volume method and gridless method at discontinuous region.