Nodal Integral Method Using Quadrilateral Elements for Transport Equations: Part 1-Convection-diffusion Equation

A generic numerical scheme for solution of the convection-diffusion equation using the nodal integral method (NIM) is developed for complex geometry. Arbitrary-shaped quadrilateral elements fitted to the complex domain are iso-parametrically mapped to unit square elements using bi-linear interpolation function. Approximations for the cross-derivative terms appearing in the transformed equations are developed and incorporated in the scheme. A numerical scheme for Neumann and mixed-type boundary conditions using NIM methodology is developed for an arbitrary-shaped boundary. Continuity conditions at the interface of two adjacent discrete cells are formulated explicitly to deal with generic quadrilateral elements. The developed scheme is verified against the analytical solutions of diffusion and convection-diffusion problems in skewed and curvilinear geometry. The results show the capability of the NIM to produce quite accurate results on a reasonably coarse mesh, even for nonorthogonal and curvilinear geometries