Building Blocks of a Numerical Solution Method

A proper model validation procedure consists of a model verification part and a part where the model predictions are compared to experimental data [61]. The model verification may be performed by the method of manufactured solutions[14 7, 163]. The method of manufactured solutions consists in proposing an analytical solution, preferably one that is infinitely differentiable and not trivially reproduced by the numerical approximation, and the produced residuals are simply treated as source terms that produce the desired or prescribed solution. These source terms or residuals are referred to as the consistent forcing functions. This method can be used to confirm that there are no programming errors in the code and to monitor the truncation error behavior during the iteration process. 

The important ingredients of a numerical solution method are outlined in this section [141, 49, 201]. 

The basis for any numerical reactor performance simulation is a mathematical model, consisting of a set of ordinary differential-, partial differential-or integro-differential equations and initial and boundary conditions. An optimized solution method must then be designed for this particular set of equations. 

The transport equations can be written in many different forms, depending on the coordinate system used. Generally, we may select the orthogonal curvilinear Cartesian-, cylindrical-, and spherical coordinate systems, or the non-orthogonal curvilinear coordinate systems, which may be fixed or moving. In reactor engineering we frequently apply the simple curvilinear 

The selected mathematical model is represented by a discretization method for approximating the differential equations by a system of algebraic equations for the variables at some set of discrete locations in space and time. Many different approaches are used in reactor engineering , but the most important of them are the simple finite difference methods (FDMs), the flrrx conservative finite volume methods (FVMs), and the accurate high order weighted residual methods (MWRs). 

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