Three Green element models for the diffusionadvection equation and their stability characteristics
Solutions of the transient 1-D diffusionadvection equation by three models of the Green element method (GEM) and their stability characteristics are presented. GEM is a novel approach of implementing the singular boundary integral theory so that computational efficiency is enhanced, and the theory is made more versatile. The first model, denoted as the quasi-steady Green element (QSGE) model, employs the Greens function of the Laplacian operator in deriving its integral representation, while the second, denoted the TGE model, uses the Greens function of the transient diffusion differential operator, and the third, denoted the ADGE model, uses the Greens function of the diffusionadvection differential operator. The first model, which had earlier been presented, is herein compared to the other two models. Three numerical examples are used to compare the accuracies of the three models. It is observed that incorporating the CrankNicholson scheme into the first model not only gives optimal results of the three models, but it more readily accommodates transport with nonuniform flow velocity field and first-order rate of decay of the pollutant. Further, the mathematical simplicity of the Greens function of the first model is an added advantage which enhances computational efficiency. The numerical stability characteristics of these models are evaluated by examining their propagation of the amplitudes and speeds of Fourier wave components in relation to their corresponding theoretical values. The results from the stability analysis confirms the superiority of the QSGE model with the CrankNicholson scheme.
diffusionadvection equation green element method boundary integral theory greens function stability analysis