Application of the Boundary Element Method Using Time Discretization to the Advection-Convection Equation

The boundary element method is a numerical computational method of solving partial differential equations which have been formulated as integral equations. It can be applied in many areas of engineering and science including fluid mechanics, acoustics, electromagnetics, and fracture mechanics. The method can be seen as a weighted residual method for solving partial differential equations, characterized by choosing an appropriate fundamental solution as a weighting function and by using the generalized Green’s formula for complete transfer of one or more partial differential operators on the weighting function. Time discretization approach requires replacing the partial derivative of the equation that involves time with a finite difference approximation, and the resulting equation now has one variable x with t becoming a constant. In this paper the advection-diffusion equation has been formulated using time discretization approach of the boundary element method. The fundamental solution of the elliptic operator has been constructed, and test examples provided.