Approximation of the Convective Transport Terms

An important aspect of Eulerian reactor models is the tmncation errors caused by the numerical approximation of the convection/advection terms [96], Very different numerical properties are built into the various schemes proposed for solving these operators. The numerical schemes chosen for a particular problem must be consistent with and reflect the actual physics represented by the model equations. 

The transient equation describing the advection of a scalar property ip yields  

The conservative form of the adveetion equation is obtained by use of the continuity equation, hence  

The truncation error associated with convection/advection schemes can be analyzed by using the modified equation method [254]. By use of Taylor series all the time derivatives except the 1. order one are replaced by space derivatives. When the modified equation is compared with the basic advection equation, the right-hand side can be recognized as the error. The presence of Ax in the leading error term indicate the order of accuracy of the scheme. The even-ordered derivatives in the error represent the diffusion error, while the odd-ordered derivatives represent the dispersion (or phase speed) error. Another method for analyzing the truncation error of advection schemes is the Fourier (or von Neumann method) [157, 158, 215]. This method is used to study the effects of numerical diffusion on the solution. 

A Taylor series analysis on the ID transport equation shows that the transient artificial viscosity coefficients for explicit upwind differencing (i.e., a scheme that is 1. order in time and space) is given by [195, 196]  

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