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

• Published in: Lobachevskii journal of mathematics Vol. 42; no. 7; pp. 1661 - 1674
• Main Authors: Hieu, L. M., Thanh, D. N. H., Tu, T. T.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01-07-2021; Springer Nature B.V
• Subjects: Algebra; Algorithms; Analysis; Approximation; Boundary conditions; Convection-diffusion equation; Differential equations; Finite difference method; Geometry; Mathematical Logic and Foundations; Mathematics; Mathematics and Statistics; Maximum principle; Probability Theory and Stochastic Processes; Regularization; convection-diffusion equation; difference maximum principle; scientific computing; non-uniform grid; regularization principle; third boundary value problem; monotone difference scheme; two-side estimate
• ISSN: 1995-0802; 1818-9962
• DOI: 10.1134/S1995080221070106

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
-norm is obtained.