Abstract
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.
Highlights
We have chosen to adopt the convection-diffusion equation for this study because of the ubiquitous role it plays in science and engineering fields
Boundary element studies related to transient convection diffusion or heat diffusion problems often portray a lack of accuracy that are not usually manifested for the steady state problems
In the work reported which points in the direction mentioned above, the commonly used boundary element formulation is hybridized
Summary
We have chosen to adopt the convection-diffusion equation for this study because of the ubiquitous role it plays in science and engineering fields. Higher order boundary element method involving free space time-dependent fundamental solutions have been used to obtain boundary integral formulation of the transient CD equation (Grigoriev and Dargush [16]) Despite these attempts, a current review of boundary element literature related to the solution of transient problems still reflects a noticeable decrease in accuracy compared to steady state problems of the same level of complexity and rigor. Domain integration oftentimes regarded as an undesirable numerical feature in BEM circles has to be addressed for these types of problems and this often gives rise to surrogate varieties of BEM formulations whose accuracy have not been fully verified for transient transport problems Competitive efforts along these lines have been recorded by Brebbia and Skerget [19] where they applied the temporal free space Green functions in two spatial dimensions to model the transport equation for low values of Peclet number. This approach totally obviates the complexities arising from the use of a two-dimensional boundary element mesh to model a one-dimensional transient problem and permits a direct focus on the numerical challenges related to the integral solutions of one-dimensional time-dependent problems
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