A modified cell-centered nodal integral scheme for the convection-diffusion equation

• Published in: Journal of computational science Vol. 80; p. 102320
• Main Authors: Ahmed, Nadeem, Singh, Suneet
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.08.2024
• Subjects: Cell-centered nodal integral method; Coarse-mesh method; Convection-diffusion equation; Differential-algebraic equations; Nodal integral method; PDE; Nodal integral method; MCCNIM; ODE; NIM; DAEs; Cell-centered nodal integral method; Coarse-mesh method; CCNIM; Differential-algebraic equations; Convection-diffusion equation
• ISSN: 1877-7503; 1877-7511
• DOI: 10.1016/j.jocs.2024.102320

The nodal integral methods (NIMs) are very efficient and accurate coarse-mesh methods for solving partial differential equations. The cell-centered NIM (CCNIM) is a simplified variant of the NIMs that has recently shown its efficiency in solving fluid flow problems but has been hampered by issues such as inapplicability to one-dimensional problems, complexities in handling Neumann boundary conditions and the formulation of a system of differential-algebraic equations (DAEs) for discrete unknowns. Here, we present a modified version of the CCNIM designed to overcome the challenges associated with its previous version. Our novel development retains the essence of CCNIM while resolving these issues. The proposed scheme, grounded in the nodal framework, achieves second-order accuracy in both spatial and temporal dimensions. Unlike its precursor, the proposed method formulates algebraic equations for discrete variables per node, eliminating the cumbersome DAE system. Neumann boundary conditions are seamlessly incorporated through a straightforward flux definition, and applicability to one-dimensional problems is now feasible. We successfully apply our approach to one and two-dimensional convection-diffusion problems with known analytical solutions to validate our approach. The simplicity and robustness of the approach lay the foundation for its seamless extension to more complex fluid flow problems. •Introducing modified cell-centered nodal integral method (MCCNIM), an advancement over the previous CCNIM approach.•Resolving intrinsic issues with the prior CCNIM system, improving computational accuracy and efficiency.•Utilization of a nodal framework for discretization in both temporal and spatial domains.•Achieving second order accuracy in spatial and temporal dimensions.•Demonstrated computational efficiency surpassing previous CCNIM, highlighting practical viability and effectiveness.