General nodal expansion method for multi-dimensional steady and transient convection-diffusion equation

A general nodal expansion method (GNEM) is developed to solve the multi-dimensional steady and transient convection-diffusion equation in this paper. The developed GNEM is an integration of the modified nodal integral method (MNIM) and nodal expansion method (NEM). Firstly, the local analytical solution of the transverse integral equation from MNIM is obtained under the framework of the basis function expansion method in NEM. Secondly, as to GNEM for time-dependent problems, the full space-time nodal method is adopted. Thirdly, GNEM borrows the nodal balance equation from NEM to ensure that the uniqueness of nodal-average variables and the conservation of all the pseudo-sources in each direction are automatically satisfied, which saves the GNEM from the complex derivation and calculations of the nodal-average variables and the zeroth order pseudo-source terms in MNIM. In this paper, GNEM has integrated the multi-dimensional transient and steady problems into the unified discrete formulation. Besides, the code based on GNEM can be easily developed from the existing NEM code since GNEM-base models have a unified formulation with conventional NEM-they only differ in the two control parameters. Meanwhile, this kind of unified formulation also makes it easier to select GNEM and conventional NEM in different computational domains for a test problem. Finally GNEM is formally developed and numerical results show that GNEM has an accuracy consistent with the MNIM, or even higher accuracy than MNIM.