On a Method of Constructing an Irregular Grid for the One-Dimensional Convection-Diffusion Equation

This paper presents a new method of constructing an irregular grid for the numerical solution of problems containing the convection-diffusion equation, which is frequently encountered in different fields of computational mathematics, physics, and chemistry. The traditional approaches use either regular grids with a large number of nodes or adaptive grids that require rearrangement at every step of solution, which may be computationally expensive. Our method is based on the transformation of an irregular grid into a regular one by using a local deformation function determined on the basis of a monotonicity criterion. This allows us to obtain a monotonic solution on a grid with a much smaller number of nodes, thereby increasing the efficiency of the difference scheme. We consider both stationary and nonstationary convection-diffusion equations, describing the corresponding grid construction algorithms for divergent and nondivergent forms of convective terms. Some examples of applying this method to various problems are given to demonstrate its advantages over the existing approaches on regular grids. The presented approach combines the advantages of irregular grids for increasing the efficiency of solution and the use of a monotonicity criterion ensuring the stability of the scheme to extend the capabilities of numerical methods for differential equations.