A local meshfree approach based on compactly supported radial basis functions for 2D coupled fluid dynamics PDEs on regular and irregular domains

Abstract

This paper presents a novel application of compactly supported radial basis functions (CSRBFs) within a local meshfree framework to solve two-dimensional coupled partial differential equations, including the Burgers’ equation (2D-CVBE) and the Saint Venant system (2D-SVS), also known as the shallow water equations. By integrating CSRBFs with the method of lines (CSRBF-MOL), this approach provides a flexible and adaptive meshfree discretization technique. It reformulates the 2D-CVBE and 2D-SVS into systems of ordinary differential equations, which are then solved numerically. The proposed method has capability to handle both rectangular and irregular domains without requiring structured grids. Its compatibility to deal with Neumann boundary conditions in makes it particularly effective for complex geometries. This framework offers a robust alternative for fluid dynamics simulations, addressing the limitations of traditional mesh-based methods in terms of flexibility and computational efficiency. The method’s accuracy and reliability are demonstrated through six numerical experiments, with results compared to global RBF approaches and conventional mesh-based techniques from the current literature.