Lagrangian meshfree finite difference particle method with variable smoothing length for solving wave equations

Abstract

In a Lagrangian meshfree particle-based method, the smoothing length determines the size of the support domain for each particle. Since the particle distribution is irregular and uneven in most cases, a fixed smoothing length sometime brings too much or insufficient neighbor particles for the weight function which reduces the numerical accuracy. In this work, a Lagrangian meshfree finite difference particle method with variable smoothing length is proposed for solving different wave equations. This pure Lagrangian method combines the generalized finite difference scheme for spatial resolution and the time integration scheme for time resolution. The new method is tested via the simple wave equation and the Burgers’ equation in Lagrangian form. These wave equations are widely used in analyzing acoustic and hydrodynamic waves. In addition, comparison with a modified smoothed particle hydrodynamics method named the corrective smoothed particle method is also presented. Numerical experiments show that two kinds of Lagrangian wave equations can be solved well. The variable smoothing length updates the support domain size appropriately and allows the finite difference particle method results to be more accurate than the constant smoothing length. To obtain the same level of accuracy, the corrective smoothed particle method needs more particles in the computation which requires more computational time than the finite difference particle method.