Nodal integral expansion method for one-dimensional time-dependent linear convection–diffusion equation

A nodal integral expansion method (NIEM) is developed as a new nodal approach and it is applied to the one-dimensional time-dependent linear convection–diffusion equation. The distinguishing feature of the new nodal scheme is to expand transverse-integrated physical quantities as well as pseudo-source terms into Legendre polynomials within a calculation node. The space-averaged physical quantity is expanded up to the first order, and the time-averaged one is expanded up to the third order, while all the pseudo-source terms are approximated to a constant. The distribution of the transverse-averaged physical quantities over a node is finally obtained by determining the expansion coefficients from the transverse-averaged ordinary differential equations, the continuity conditions of the transverse-averaged physical quantities and their derivatives, and other constraint equations. Calculations for sample problems are carried out to test the effectiveness of the scheme, and it is observed that the NIEM scheme has capability to yield highly accurate solutions and to track the evolution of the physical quantities very well. In addition, it can be reasonably concluded that the accuracy of the NIEM is comparable on the whole to the existing nodal method of the modified nodal integral method (MNIM).