A novel direct interpolation boundary element method formulation for solving diffusive–advective problems

The capability of dealing with the advective transport term constitutes a challenging issue for the performance of the majority of the numerical techniques, which significantly lose their precision with the increasing in the relative magnitude of this term. Recently, a new boundary element technique called direct interpolation (DIBEM) has emerged, mainly characterized by the approximation of the entire kernel of the remaining domain integrals. The DIBEM model has been successfully applied to several scalar field problems and recently applied to certain diffusive–advective problems with superior accuracy compared to dual reciprocity formulation (DRBEM). Using DIBEM, a broader range of precise responses for flows with higher Peclet numbers can be reached beyond a lower computational cost due to the simplicity of its matrix operations. These advantages are due DIBEM concept that employs a simple interpolation procedure to approximate the kernel of the advective domain integral. However, it was noticed that in some physical scenarios with spatial variation in the velocity field, the accuracy of DIBEM did not have the same quality observed in other applications. Therefore, this work presents a new DIBEM model so that the integral equations include the explicit calculation of spatial derivatives and simultaneously solve the variation of the divergent velocity field if this is not zero. Conversely, reactive and source terms were also included to expand numerical comparisons. Thus, in the proposed examples, the results of two DIBEM models, the standard (DIBEM-S) and the new, so-called alternative model (DIBEM-A), are compared with DRBEM and also with available analytical solutions.