Surface only integral equation approach for convection-diffusion problems with variable velocity

Mariela Castillo has recently successfully passed her PhD Viva at the Wessex Institute of Technology for a thesis dealing with the application of a new fundamental solution in fluid mechanics. ("Surface only integral equation approach for convection-diffusion problems with variable velocity").

Mariela is a graduate in mathematical sciences from the Universidad Central in Caracas where, upon successful completion of her PhD at WIT, she has been appointed as Assistant Professor. The Universidad Central is internationally renowned for its excellent academic standards and is one of the best schools in Latin America.

The external examiner of the thesis was Prof Alan Davies of Hertfordshire University, with Dr Viktor Popov, Head of Environmental and Fluid Mechanics at Wessex Institute acting as the Internal Examiner. Mariela was congratulated by the examiners for the quality of her thesis and her excellent presentation.

Mariela's work consists of the development of a new surface only BEM numerical scheme to solve convection diffusion problems with variable velocity field. A substantial number of numerical models for the convection-diffusion equation have appeared previously in the literature. Most of these models employ either the finite difference (FDM) or the finite element (FEM) methods of solution, and give emphasis to algorithms to suppress the well-known problems of spurious oscillations and damping of wave fronts intrinsic to these methods.

The boundary element method (BEM) is now a well-established numerical technique in fluid mechanics. The general solution of the convection-diffusion equation with variable velocity can be given in terms of a surface integral representation formula, which can be obtained from the corresponding Green's second identity and the use of the fundamental solution of the adjoint partial differential equation. However, the use of the BEM in terms of this integral representation formula has been mainly limited to the simple cases of constant velocity field, for which the closed form of the fundamental solution of the adjoint partial differential equation is known. Although it is theoretically possible to guarantee the existence of the fundamental solution for the case of variable velocity, its expression is only known for simple cases.

BEM formulations for treating problems with variable velocity fields have employed the fundamental solution of Laplace's equation and treated the convective terms as pseudo-sources. The application of this approach yields an integral representation formula of the surface and boundary type, where the pseudo-sources terms appear as volume potential. In these cases the BEM is computationally more costly than classical domain schemes.

Mariela instead has developed a new technique to find the fundamental solution of the adjoint equation to the convection-diffusion equation with variable velocity in terms of the fundamental solution of the Laplace equation plus a regular function, which can be obtained by the unique solution of a Volterra integral equation of the second kind.

Two different analytical approaches to solving the resulting Volterra integral equation based upon series representations have been considered, besides an alternative numerical approach based on the same representation is presented.

An alternative interesting approach to finding non-transient fundamental solution when the close form of the transient fundamental solution is known, is obtained by the asymptotic limit of a numerical time integration, in which the numerical singularity of the kernel is removed by adding and subtracting the fundamental solution of the Laplace equation in its integral and close form.

Using the expressions obtained for the fundamental solution, several numerical examples were considered based upon the corresponding BEM formulation of the convection-diffusion problem with variable velocity field in terms of such fundamental solution. Obtaining in this way a new BEM numerical scheme, which is free from the need of domain integrations (surface only approach) and consequently is not as computationally costly as those previous surface domain BEM approaches and also is free from any spurious oscillations, as is the case of the traditional FDM and FEM approaches. The examples tested with this new approach have demonstrated the efficiency and flexibility of the proposed new numerical scheme.