Smoothed particle hydrodynamics (SPH) offers distinct advantages for modeling many engineering problems, yet achieving high-order consistency in its conservative formulation remains to be addressed. While zero- and higher-order consistencies can be obtained using particle-pair differences and kernel gradient correction (KGC) approaches, respectively, for SPH gradient approximations, their applicability for discretizing conservation laws in practical simulations is limited due to their lack of discrete conservation. Although the standard anti-symmetric SPH approximation is able to achieve conservation and zero-order consistency through particle relaxation, its straightforward extensions with the KGC fail to satisfy zero- or higher-order consistency. In this paper, we propose the reverse KGC (RKGC) formulation, which is conservative and able to satisfy up to first-order consistency when particles are relaxed based on the KGC matrix. Extensive numerical tests show that the new formulation considerably improves the accuracy of the Lagrangian SPH method. In particular, it is able to resolve the long-standing high-dissipation issue for simulating free-surface flows. Furthermore, with fully relaxed particles, it enhances the accuracy of the Eulerian SPH method even when the ratio between the smoothing length and the particle spacing is considerably reduced. The reverse KGC formulation holds the potential for extension to even higher-order consistencies with a pending challenge in addressing the corresponding particle relaxation problem.

中文翻译：

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实现保守 SPH 近似的高阶一致性和收敛


平滑粒子流体动力学 （SPH） 为许多工程问题建模提供了明显的优势，但在其保守公式中实现高阶一致性仍有待解决。虽然对于 SPH 梯度近似，可以分别使用粒子对差分和核梯度校正 （KGC） 方法获得零阶和高阶一致性，但由于缺乏离散守恒，它们在实际模拟中离散守恒定律的适用性受到限制。尽管标准的反对称 SPH 近似能够通过粒子弛豫实现守恒和零阶一致性，但其与 KGC 的直接扩展无法满足零级或更高阶一致性。在本文中，我们提出了反向 KGC （RKGC） 公式，该公式是保守的，并且在基于 KGC 矩阵的颗粒松弛时能够满足一阶一致性。广泛的数值测试表明，新公式大大提高了拉格朗日 SPH 方法的准确性。特别是，它能够解决模拟自由表面流时长期存在的高耗散问题。此外，对于完全松弛的颗粒，即使平滑长度与颗粒间距之间的比率大大降低，它也能提高欧拉 SPH 方法的准确性。反向 KGC 公式有可能扩展到更高阶的一致性，在解决相应的粒子弛豫问题方面存在悬而未决的挑战。