Integral equation formulation of an unsteady diffusion–convection equation with variable coefficient and velocity

In this paper we present an integral equation formulation for the time dependent diffusion–convection equation with variable coefficient and velocity with sources. The formulation is based on usage of the steady fundamental solution of the convection–diffusion equation. For a known velocity and coefficient fields, which may change with location and time, the formulation avoids the usage of the gradient of the unknown field function and thus avoids making the problem nonlinear. Two discretization approaches are proposed and compared: a standard single domain boundary–domain element technique and a domain decomposition approach. The validity of the formulation and comparison of discretization approaches is preformed on several challenging test cases. Mesh convergence is reported and the advantages and disadvantages of both approaches are examined.