In this study, we focus on space-time mesh-free numerical techniques for efficiently solving the Richards equation which is often used to model unsaturated flow through porous media. We propose an efficient approach which combines the use of local multiquadric (MQ) radial basis function (RBF) methods and space-time techniques. The localized MQ-RBFs meshless methods allow to avoid mesh generation and ill-conditioning problem where a sparse matrix is obtained for the global system which has the advantage of using reduced memory and computational time. To further reduce the computational cost, we use the space-time techniques having the advantages of solving the resulting algebraic system only once and removing the time-integration procedure. The proposed method has the benefit of considering collocation points on the boundaries of computational domains which makes it more flexible in dealing with complex geometries. We implement the proposed numerical model of infiltration and we perform a series of numerical tests, encompassing various nontrivial solutions, to confirm the performance of the proposed techniques. The numerical simulations show the accuracy, efficiency in terms of computational cost, and capability of the proposed numerical techniques in solving the Richards equation in two-, three- and four-dimensional space-time domains with complex boundaries.

在这项研究中，我们专注于有效求解理查兹方程的时空无网格数值技术，该方程常用于模拟通过多孔介质的非饱和流动。我们提出了一种有效的方法，它结合了局部多二次 (MQ) 径向基函数 (RBF) 方法和时空技术的使用。局部 MQ-RBFs 无网格方法允许避免网格生成和病态问题，其中为全局系统获得稀疏矩阵，其优点是使用减少的内存和计算时间。为了进一步降低计算成本，我们使用时空技术，其优点是只需求解所得代数系统一次，并去除时间积分过程。所提出的方法的好处是考虑了计算域边界上的搭配点，这使得它在处理复杂的几何图形时更加灵活。我们实施了所提出的渗透数值模型，并进行了一系列数值测试，包括各种非平凡的解决方案，以确认所提出技术的性能。数值模拟显示了所提出的数值技术在求解具有复杂边界的二维、三维和四维时空域中的理查兹方程的准确性、计算成本方面的效率和能力。以确认所提出技术的性能。数值模拟显示了所提出的数值技术在求解具有复杂边界的二维、三维和四维时空域中的理查兹方程的准确性、计算成本方面的效率和能力。以确认所提出技术的性能。数值模拟显示了所提出的数值技术在求解具有复杂边界的二维、三维和四维时空域中的理查兹方程的准确性、计算成本方面的效率和能力。