Geometric Conservation Law in ALE Formulations

Abstract

The aim of this work is to study the influence of the Geometric Conservation Law (GCL) when numerical simulations are performed on deforming domains with an Arbitrary Lagrangian-Eulerian (ALE) formulation. This analysis is carried out in the context of the Finite Element Method (FEM) for the scalar advection-diffusion equation defined on a moving domain. Solving the problem on a moving mesh using an ALE formulation needs the computation of some geometric quantities, such as element volumes and Jacobians, which involve the nodal positions and velocities. The so-called Geometric Conservation Law (GCL) is satisfied if the algorithm can exactly reproduce a constant solution on moving grids. Not complying with the GCL means that the stability of the time integration is not assured and, thus, the order of convergence could not be preserved. To emphasize the importance of fulfilling the GCL, numerical experiments are performed in 2D using several mesh movements. In these experiments different temporal integration schemes have been used .