A gradient reproducing kernel based stabilized collocation method (GRKSCM) to numerically solve complicated nonlinear Korteweg-de Vries (KdV) equation is proposed in this paper. The acquisition of GRK through high-order consistency conditions reduces the complexity of RK derivative operations and remarkably improves the effectiveness of the proposed method by directly forming GRK approximations. Owing to the fulfillment of high-order integration constraints, the SCM achieves exact integration in the domain and on boundaries. Von Neumann analysis is utilized to establish the stability criteria for GRKSCM when combined with forward difference temporal discretization. The effectiveness of the GRKSCM method in solving the KdV equation is investigated through six numerical examples. The examples include the motion of the single solitary wave propagation, interaction between two solitary waves and interaction among three solitary waves. Furthermore, the behavior of a two-dimensional solitary wave and the propagation of a three-dimensional solitary wave are also numerically investigated. The numerical outcomes confirm that GRKSCM provides high accuracy in comparison with analytical solutions. In addition, the invariants and error analysis show the conservation of our proposed GRKSCM method.

中文翻译：

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使用稳定配置法和梯度再生核近似的非线性KdV方程的无网格方法


本文提出了一种基于梯度再生核的稳定配置方法（GRKSCM）来数值求解复杂的非线性Korteweg-de Vries（KdV）方程。通过高阶一致性条件获取GRK，降低了RK导数运算的复杂度，并通过直接形成GRK近似，显着提高了该方法的有效性。由于满足高阶集成约束，SCM实现了域内和边界上的精确集成。冯诺依曼分析用于建立 GRKSCM 与前向差分时间离散相结合的稳定性标准。通过六个数值算例研究了 GRKSCM 方法求解 KdV 方程的有效性。这些例子包括单个孤立波传播的运动、两个孤立波之间的相互作用以及三个孤立波之间的相互作用。此外，还对二维孤立波的行为和三维孤立波的传播进行了数值研究。数值结果证实 GRKSCM 与解析解相比具有较高的准确性。此外，不变量和误差分析表明了我们提出的 GRKSCM 方法的守恒性。