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.

使用节点积分法 (NIM) 求解偏微分方程 (PDE) 的主要优点是它们可以在相对粗糙的网格上获得准确的解。由于在推导中使用了横向积分程序，大多数开发的 NIM 仅限于具有各向同性扩散项的 PDE。最近，NIM 已使用等参映射方法扩展到任意几何形状，这导致了各向同性扩散到各向异性扩散的转换。这需要开发一种方法来近似与传统 NIM 的准确性顺序一致的交叉导数项。在本文中，使用 NIM 对 3D、时间相关的各向异性对流扩散方程进行了数值求解。开发了一种基于 NIM 的实际离散未知量（即线平均或表面平均变量）来近似交叉导数项的新方法。此外，先前开发的基于角点值的 2D 中交叉导数项的近似值（Kumar 等人，2013 年）已扩展到 3D。解决了六个数值测试用例来测试两种方法的准确性和效率。即使对于两种交叉导数近似的粗网格，各向异性扩散方程的 NIM 精度也保持不变。然而，在所有情况下，基于表面平均的近似比基于点值的近似更准确和有效。使用基于表面平均的近似值的 NIM 在空间和时间上是二阶精确的。另一方面，