Second Order Monotone Finite-Difference Schemes on Non-Uniform Grids for Multi-Dimensional Convection-Diffusion Problem with a Boundary Condition of the Third Kind

In this article, we present a study on constructing a second order local approximation monotone difference schemes on spatial non-uniform grids for the parabolic equation of convection-diffusion type with a third kind boundary condition without using the basic differential equation at the boundary of the domain. The goal is a combination of the differential inequality, the regularization principle and the assumption of the existence and uniqueness of a smooth solution. In this case, the boundary conditions are directly approximated with the second order on a two-point stencil. The convergence of the proposed algorithms to the solution of the original differential problem with the second order is proved. With the help of difference maximum principle, two-sided estimates of the difference solution are established and an important a priori estimate in a uniform $$C$$ -norm is obtained.