Substantial part of my contribution to research and development is in the field of numerical methods for solving inverse and ill-posed problems. Method of Variational Imbedding (MVI) is a new approach to devising robust difference schemes and algorithms for numerical solution of inverse or ill-posed problems proposed by Prof. Christov. MVI was successfully applied for identification of the coefficient in
parabolic equation; coefficient identification in
diffusion equation from over-posed data at the boundary
[
Jour.:98T,
Jour.:99T ]; identification of the
boundary layer thickness as an inverse problem for coefficient identification developed in
[
Jour.:97BLT
Proc:00BLT,
Jour.:96BLT, ].
As it is seen from my curriculum vitae I have special interests in construction of computationally efficient schemes (particularly splitting schemes) for solving nonlinear problems governed by partial differential equations. The splitting procedure reduces in order of magnitude the number of operations per iteration comparing with application of direct solvers. The latter requires large memories and it is not feasible for large scale computations, particularly, for three dimensional problems: splitting or multigrid are very efficient techniques for solution of nonlinear elliptic systems.
My present research program includes developing methods for Navier-Stokes and advection-diffusion equations. Currently, new finite element time splitting scheme for Navier-Stokes equations in general coordinates is being developed in collaboration with Dr. Peter Minev from University of Alberta. The method is based on modified pressure Poisson equation employed in [ Proc.:00NS1 ].
The fully implicit coordinate splitting has been developed for the Navier-Stokes equation in primitive variables. The method allows for complete coupling of the boundary conditions. Conservative approximations for the advective terms are employed on an uniform staggered grids. The proposed difference scheme and algorithm are robust for large values of Reynolds number Re and it has been elaborated to 3D case and unsteady Navier-Stokes equations. Some results are presented in [ Jour.:03NS,
Proc.:00NS2,
Proc.:00NS ]. A bicyclic splitting finite difference scheme, which is applicable for multi-dimensional problems, is proposed in
[ Jour.:AD02,
Proc.:AD00 ]
for solving the unsteady advection-diffusion problem, while in
[ Jour.:AD101,
Jour.:AD201 ]
a conservative numerical scheme based on characteristic line method has been investigated. A difference scheme for solving a simple mathematical model of the opposite currents (the stationary boundary layer approximation for two immiscible flows) is proposed in [ Jour.:BL03 ]. The governing equations for each flow are of parabolic type but the type of the whole system is non-parabolic because of the conditions of conjugation for tangential velocities and forces.
