A fourth-order accurate quasi-variable mesh compact finite-difference scheme for two-space dimensional convection-diffusion problems

Abstract

We discuss a new nine-point fourth-order and five-point second-order accurate finite-difference scheme for the numerical solution of two-space dimensional convection-diffusion problems. The compact operators are defined on a quasi-variable mesh network with the same order and accuracy as obtained by the central difference and averaging operators on uniform meshes. Subsequently, a high-order difference scheme is developed to get the numerical accuracy of order four on quasi-variable meshes as well as on uniform meshes. The error analysis of the fourth-order compact scheme is described in detail by means of matrix analysis. Some examples related with convection-diffusion equations are provided to present performance and robustness of the proposed scheme. MSC: 65N06; 65N12; 35J57; 35J72

A high-resolution fuzzy transform combined compact scheme for 2D nonlinear elliptic partial differential equations

“…In the two-dimensional space, a significant drawback of the conventional approach is the blind expansion of the same computational scheme independently on the order of approximation; hence, the computational complexity increases. Second-order approximations lead to low accuracy of the solution and are inapplicable to PDE boundary conditions with singularities [14 , 15] . As a result, numerical schemes formulated on high-order approximations with a minimum number of mesh points in a single cell network are more appropriate [16] .…”

“…With these assumptions, the nonlinear elliptic PDEs (1) possess a unique solution [18] . Such types of PDEs in two and three-dimension by compact finite-difference approximations are described in [14 , 18 , 19] . We elaborate a high-resolution mechanism to design an algorithm by going through -transform combined with compact nine-point discretization that offers fourth-order solution accuracy.…”

Method of approximations for the convection-dominated anomalous diffusion equation in a rectangular plate using high-resolution compact discretization

“…One needs to assign the boundary data (2) for the algorithmic implementation, and the iterative method helps determine approximate solution values. The computational scheme for regular elliptic PDEs in two dimensions has been obtained in the past [ 27 , 28 ].…”

A fourth-order arithmetic average compact finite-difference method for nonlinear singular elliptic PDEs on a 3D smooth quasi-variable grid network

“…Three-point compact formulation to convection and diffusion terms by considering quasi-variable grids are described in the literature [17 , 18] . Digital electrochemistry, wind-driven ocean circulation, and convection-diffusion phenomenon are some of the real-world application models that use quasi-variable grid topology for high-order discretization in PDEs possessing interior or boundary layers [18] , [19] , [20] , [21] , [22] .…”