Nodal integral methods in general 2D curvilinear coordinates - applied to convection–diffusion equation in domains discretized using quadrilateral elements

Nodal integral method (NIM) is derived in general, 2D, curvilinear coordinates and applied to solve the convection–diffusion equation in domains discretized using quadrilateral elements. The quadrilateral elements in the Cartesian system are transformed into rectangular elements in curvilinear coordinates using bi–linear Lagrangian interpolation functions. An approximation for the transverse-integration operator is developed in curvilinear coordinates. The convection–diffusion equation and the continuity conditions at the common edge between two adjacent nodes are derived in curvilinear coordinates. A new approximation of the cross–derivative terms resulting from transforming the governing equation to curvilinear coordinates is developed using the discrete unknowns of NIM (line–averaged variables). Three numerical test problems are solved to assess the accuracy and efficiency of the newly developed scheme. The results show that the scheme developed in this paper is second–order accurate in space and time, which is the same order of accuracy for traditional NIM for regularly-shaped elements. The results show that the accuracy of NIM for quadrilateral elements is maintained even for coarse mesh and highly distorted elements.