An arbitrary Lagrangian discrete least squares meshfree method for discontinuity capturing with application in the Saint–Venant equations

A discrete least squares meshfree (DLSM) method has been developed to solve the partial differential equations (PDEs) governing physical phenomena, particularly in the field of fluid mechanics. Existing DLSM methods are restricted to the solution of either the fully Eulerian or Lagrangian form of the governing PDEs. Despite its successful implementation for various PDEs, Eulerian DLSM has limitations due to problems with moving boundaries. Considering the mesh-independent flexibility of DLSM, this issue can be addressed with a Lagrangian DLSM. However, fully Lagrangian DLSM may suffer from numerical issues due to extreme nodal clustering or dispersion. Additionally, both existing fully Eulerian and Lagrangian DLSM methods are prone to the spurious oscillations around discontinuities and steep gradients. In this paper, an arbitrary Lagrangian DLSM (AL-DLSM) method is proposed to improve the accuracy of the method by addressing this issue. The Lagrangian governing PDEs are solved followed by a Lagrangian approach for rearranging nodal positions in a deforming domain. In the present approach, an entropy-based arbitrary nodal reconfiguration is proposed to refine nodal distributions in high-gradient (high entropy density) regions, effectively addressing the issue of non-physical oscillations around discontinuities. The proposed AL-DLSM method is applied to solve the shallow water equations, namely the Saint–Venant equations, and its accuracy and effectiveness are verified. The results demonstrate that the proposed AL-DLSM method is considerably more accurate compared to the existing fully Eulerian or fully Lagrangian forms of the DLSM method, particularly for problems involving steep gradients.