On the numerical solution to the truncated discrete SPH formulation of the hydrostatic problem

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 C2. 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.