## 4 Method for Assembling Numerical Schemes
### Summary  
We may now condense the various aspects of choosing appropriate numerical schemes as follows into a unifying method, given the restrictions we posed in Section 4.1.  
First, we take three sets of inputs that are of practical relevance: the mathematically formulated, continuous problem, the computational domain that one wishes to solve the former on, and the configuration of the target hardware.  
To design the intended decision process, we start by evaluating the decision metrics that impact the target scheme in the most general manner at first. The general question of whether the prescribed system of PDEs requires an efficient solution on a large scale fulfills this requirement here. By the term large scale, we understand state-of-the-art computing hardware on massively parallel architectures. That decision, in turn, is influenced by two factors: One may directly intend to efficiently solve the system of PDEs on that hardware, or the multiscale nature of the problem demands such a computing environment. If either is the case, solving the entire system using the HDG method is advisable due to the resulting locality of the problem.  
The remaining parts of the decision process depend on the class of PDE present. From here on, we operate in a field-wise manner and classify the system of PDEs for each independent variable separately. If a PDE is convective in character, that is, hyperbolic, we recommend the use of numerical schemes that incorporate discontinuous approximations. But, if a problem is diffusive by nature, the solution will be continuous and, thus, the use of continuous approximations is more advisable.  
In the case of the former, following the discussion in Section 4.4, a final disambiguation must be made regarding linearity. If the PDE is linear or semilinear, a DG scheme can be applied due to the unlikeliness of stability issues. Otherwise, the use of a simple FV scheme is more advisable to obtain a stable solution without having to iterate through many different choices of flux limiters in a trial-and-error fashion.  
Regarding the continuous schemes, as was explained in Sections 3.3 and 4.3, the configuration of the domain geometry plays an important role in the efficiency of the overall scheme. If the domain is Cartesian, irrespective of dimensionality, the FDM can deliver accurate results with a considerably decreased number of arithmetic operations. The conceptual flexibility of the FEM regarding the domain is then unnecessary. In the other case, though, where the domain is topologically more complex, relying on FEM algorithms that account for the necessary global mappings is more appropriate. It would, of course, be possible to identify a middle ground between both schemes, for example, when a simple and prescribed transformation can be applied to the entire domain. This would, for example, be the case for systems that can be described by polar coordinates. However, few computer codes implement such functionality. As the focus of this method lies on practicality and usefulness, we rather choose a method that can make use of widespread and established computer codes and thus omit these possibilities.  
As a result, we obtain one process that guides the user through iteratively selecting the most appropriate combination of numerical schemes for a given, fixed, and well-defined set of inputs. This method may be summarized in a flow chart, which is depicted in Figure 5.