Properties of Discretization Schemes

A proper convergence criterion is important, from both the accuracy and efficiency points of view, because it is deciding when to stop the iterative process. Research codes are generally iterating until the machine accuracy is reached, whereas the commercial codes are less accurate as efficiency is commonly desired by the customers. In commercial CFD codes, a convergence criterion defined by the reduction of the normalized residual, as calculated from the initial guess variable values, by a factor of 10 is frequently considered sufficient by contract research- and salespersons. However, for complex multiphase reactive flows this approach may easily lead to unphysical solutions. 

For most multiphase reactive flow problems, it is not possible to analyze all the operators in the complete solution method simultaneously. Instead the different operators of the method are analyzed separately one by one. The working hypothesis is that if the operators do not possess the desired properties solely, neither will the complete method. Unfortunately, the reverse is not necessarily true. In practical calculation we can only use a finite grid resolution, and the numerical results will only be physically realistic when the discretization schemes have certain fundamental properties. The usual numerical terminology employed in the CFD literature is outlined in this section [141, 202, 49]. 

A consistent numerical scheme produces a system of algebraic equations which can be shown to be equivalent to the original model equations as the grid spacing tends to zero. The truncation error represents the difference between the discretized equation and the exact one. For low order finite difference methods the error is usually estimated by replacing all the nodal values in the discrete approximation by a Taylor series expansion about a single point. As a result one recovers the original differential equation plus a remainder, which represents the truncation error. 

A numerical solution method is said to be stable if the method does not magnify the errors that appear during the numerical solution process. This property is relevant as a consistent discretization scheme provides no guarantee that the solution of the discretized equation system will become an accurate solution of the differential equation in the limit of small step size. The stability of low order numerical schemes applied to idealized problems can be analyzed by the von Neumann s method. However, when solving relevant, non-linear and coupled reactor model equations with complex boundary 

A matrix which is n x m with k non-zero entries is sparse if fc C n X m. 

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