The aim of this work is to study the solution of the smoothed particle hydrodynamics (SPH) discrete formulation of the hydrostatic problem with a free surface. This problem, in which no time dependency is considered, takes the form of a system of linear equations. In particular, the problem in one dimension is addressed. The focus is set on the convergence when both the particle spacing and the smoothing length tend to zero by keeping constant their ratio. Values of this ratio of the order of one, corresponding to a limited number of neighbors, are of practical interest. First, the problem in which each particle has one single neighbor at each side is studied. The explicit expressions of the numerical solution and the quadratic error are provided in this case. The expression of the quadratic error demonstrates that the SPH solution does not converge to the exact one in general under the specified conditions. In this case, the error converges to a residue, which is in general large compared to the norm of the exact solution. The cases with two and three neighbors are also studied. An analytical study in the case of two neighbors is performed, showing how the kernel influences the accuracy of the solution through modifying the condition number of the referred system of linear equations. In addition to that, a numerical investigation is carried out using several Wendland kernel formulas. When two and three neighbors are involved it is found that the error tends in most cases to a small limiting value, different from zero, while divergent solutions are also found in the case of two neighbors with the Wendland Kernel C². Other cases with more neighbors are also considered. In general, the Wendland Kernel C^2. turns out to be the worst choice, as the solution is divergent for certain values of the ratio between the particle spacing and the smoothing length, associated with an ill-conditioned matrix.

中文翻译：

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关于静水力问题截断离散SPH公式的数值解

这项工作的目的是研究具有自由表面的静水问题的光滑粒子流体动力学（SPH）离散公式的解决方案。这个不考虑时间依赖性的问题采取线性方程组的形式。特别地，解决了一维问题。当粒子间距和平滑长度都通过保持恒定的比率趋于零时，将焦点放在会聚上。具有一定数量级的邻居的该比例的比率值具有实际意义。首先，研究每个粒子在每一侧都只有一个相邻粒子的问题。在这种情况下，提供了数值解和二次误差的显式表达式。二次误差的表达式表明，在指定条件下，SPH解通常不会收敛到精确的解。在这种情况下，误差收敛为残差，与精确解的范数相比，残差通常较大。还研究了具有两个和三个邻居的情况。在两个邻居的情况下进行了分析研究，显示了内核如何通过修改线性方程组的条件数来影响解的精度。除此之外，使用几种Wendland核公式进行了数值研究。当涉及到两个和三个邻居时，发现在大多数情况下，误差趋向于一个小的极限值，该极限值不同于零，C ²。还考虑了其​​他邻居更多的情况。通常，Wendland Kernel C ²最终是最糟糕的选择，因为对于病态矩阵，粒子间距和平滑长度之间的比率的某些值的解是不同的。