Fast boundary-domain integral method for unsteady convection-diffusion equation with variable diffusivity using the modified Helmholtz fundamental solution

In this paper, we develop a boundary-domain integral formulation of the unsteady convection-diffusion equation with variable material properties. The derivation is based on the Green's second theorem using the fundamental solution of the modified Helmholtz equation. Several discretisation approaches are considered: the full matrix and domain-decomposition approaches are compared with adaptive cross-approximation and wavelet-based approximation techniques. With the use of modified Helmholtz fundamental solution, whose shape is determined by the time step size and diffusivity, we are able to achieve an improvement in the final approximated matrix size. We present several numerical tests to verify the validity of the proposed integral formulation and assess the approximation properties for different diffusivity variations and different Peclet numbers. We develop guidelines for choosing the user prescribed parameters such as the hierarchical matrix admissibility parameter, the adaptive cross-approximation rank determination parameter and the wavelet thresholding parameter.