A new corrective scheme for the derivative-free (DF) smoothed particle hydrodynamics (SPH) method is proposed to compute the first- and second-order derivatives. To demonstrate the ability of the new corrective scheme, the first- and second-order derivatives of the given function were computed when the particles were uniformly and irregularly distributed. The nonlinear convection–diffusion problems were simulated using both the DF-SPH method and the proposed method in one- and two-dimensional cases, and root mean square errors were compared. Results show that numerical solutions acquired by the proposed corrective scheme agree very well with analytical solutions and that the new scheme is more efficient, accurate, and stable than the original DF-SPH.

中文翻译：

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DF-SPH的新纠正方案

提出了一种新的无导数（DF）平滑粒子流体动力学（SPH）方法的校正方案，以计算一阶和二阶导数。为了证明新的纠正方案的能力，当粒子均匀且不规则地分布时，计算给定函数的一阶和二阶导数。在一维和二维情况下，使用DF-SPH方法和所提出的方法对非线性对流扩散问题进行了模拟，并比较了均方根误差。结果表明，所提出的校正方案获得的数值解与解析解非常吻合，并且新方案比原始的DF-SPH更有效，准确和稳定。