Boundary element methods for transient convective diffusion. Part III: Numerical examples

Abstract

Higher-order boundary element methods (BEM) for transient convective diffusion phenomena presented in Parts I and II of the paper are implemented numerically to examine their performance for a series of model problems. In order to highlight the importance of proper resolution in time and space for the transient problems, we consider both the one- and two-dimensional formulations in this paper. We utilize an implicit time-recurring formulation to advance in the time domain. The use of higher-order time interpolation functions facilitates extremely high resolution with respect to time step sizes, and thus necessitates similar levels of spatial resolution. In this paper, linear, quadratic and quartic boundary elements as well as bi-linear, bi-quadratic and bi-quartic volume cells are implemented to ensure a desired high level of accuracy both in time and in space. In order to investigate the performance of the BEM formulations, we introduce five problems of unsteady convection–diffusion that possess exact solutions. For all five numerical examples considered in this paper, the higher-order BEM demonstrate an extremely high level of accuracy even for predominantly convective flows. It is shown that the use of the convective kernels provide an analytical upwinding for any Peclet number and any mesh orientation with respect to the flow direction. The introduction of the kernel influence domains facilitates a very efficient and robust algorithm for integration over boundary elements and volume cells. Due to the nature of the convective kernels, the resulting global matrix is sparse with the non-zero elements localized around the main diagonal. The preconditioned bi-conjugate gradient method utilized in the paper allows very efficient factorization of the global sparse matrix. Moreover, the efficiency of the BEM formulations increases dramatically with the increase of the Peclet number of the flow.