Nodal integral method for 3D time-dependent anisotropic convection-diffusion equation

Abstract

The main advantage of using Nodal Integral Methods (NIM) in solving partial differential equations (PDEs) is that they lead to an accurate solution over relatively coarse meshes. Due to the use of the transverse integration procedure in the derivation, most of the NIMs developed were restricted to PDEs with isotropic diffusion terms only. Recently, the NIM has been extended to arbitrary geometries using iso-parametric mapping approach, which resulted in the transformation of isotropic diffusion to anisotropic diffusion. This required the development of a method to approximate the cross-derivative terms that is consistent with the order of accuracy of traditional NIMs. In this paper, the 3D, time-dependent, anisotropic convection-diffusion equation is solved numerically using the NIM. A new method to approximate the cross-derivative terms based on the actual discrete unknowns of the NIM – namely, the line-averaged or surface-averaged variables – is developed. Also, the previously developed approximation for the cross-derivative terms in 2D (Kumar et al., 2013) that is based on the corner point values is extended to 3D. Six numerical test cases are solved to test the accuracy and efficiency of both approaches. The accuracy of NIM for the anisotropic diffusion equation is maintained even for coarse meshes for both cross-derivative approximations. However, the surface-averaged-based approximation is more accurate and efficient than the point-value-based approximation in all cases. The NIM using the surface-averaged-based approximation is second-order accurate in space and time. On the other hand, the NIM using the point-value-based approximation is between first and second-order accurate in space, and second-order accurate in time.