The Effect of Time-Stepping on the Accuracy of Green Element Formulation of Unsteady Convective Transport

Despite the significant number of boundary element method (BEM) solutions of time-dependent problems, certain concerns still need to be addressed. Foremost among these is the impact of different time discretization schemes on the accuracy of BEM modeling. Although very accurate for steady-state problems, the boundary element methods more often than not are computationally challenged when applied to transient problems. For the work reported herein, we investigate the level of accuracy achieved with different time-discretization schemes for the Green element method (GEM) solution of the unsteady convective transport equation. The Green element method (a modified BEM formulation) solves the boundary integral theory (A Fredholm integral equation of the second kind) on a generic element of the problem domain in a way that is typical of the finite element method (FEM). In this integration process a new system of discrete equations is produced which is banded and hence amenable to matrix manipulations. This is subsequently deployed to investigate the proper resolution in both space and time for the chosen transient 1D transport problems especially those involving shock wave propagation and different types of boundary conditions. It is found that for three out of the four numerical models developed in this study, the new system of discrete element equations generated for both space and temporal domains exhibits accurate characteristics even for cases involving advection-dominant transport. And for all the cases considered, the overall performance relies heavily on the temporal discretization scheme adopted.