## 3 Look-Up Table Generation and Testing
### Table Density/Accuracy Investigation  
Before applying the CPV solver to engine simulations in the SRM, the interpolation strategy for the progress variable source term retrieval was verified. For each of the data entry points included in the tabulation grid (see Table 1), the auto-ignition solution from the constant pressure reactor calculation using the online chemistry solver has been compared to the respective solution retrieved from the table using linear interpolation. The discrepancy between the two solution has been quantified based on the mean relative difference of the vector composed by the errors at 5, 10, 50 and 90 percent of energy released location (noted as E\({}_{5\%}\), E\({}_{10\%}\), E\({}_{50\%}\) and E\({}_{90\%}\) in Figure 4) at all given reactor points.  
The global interpolation error has been then defined based on the mean square difference (MSD) of the location of E\({}_{5\%}\), E\({}_{10\%}\), E\({}_{50\%}\) and E\({}_{90\%}\) points. In Equations (7), where i = 5, 10, 50, 90, the exact definition of the interpolation error is reported.  
\[\varepsilon_{\rm i}=\frac{\rm E_{i,T,D\rm tabulated}-E_{i,Online}}{\rm E_{i,Online }}\times 100\,\overline{\varepsilon}=\sum_{\rm i=1}^{\rm N_{errors}}|\varepsilon_{ \rm i}|/\rm N_{errors}\ \varepsilon_{\rm MSD}=\sqrt{\sum_{\rm i}(\overline{ \varepsilon}-|\varepsilon_{\rm i}|)^{2}} \tag{7}\]  
The interpolation errors have been assessed for a large set of operating conditions covering lean to rich (\(\phi\) = 0.6\(-\)3.0) conditions as well as from low to high unburned temperatures (T = 600-1400 K) under constant pressure reactor conditions. \(n\)-Heptane was used as fuel molecule, and the mechanism developed by [PERSON] et al. (2018) was employed. The main goal of this campaign was to quantify the accuracy of the interpolation routine under the simplest reactor conditions, so to have a higher confidence in results assessment during the subsequent engine simulation campaign. In the SRM simulation, numerous sub-models are coming into play, and hence, error compensation effects may arise. In Figure 5, exemplary results of the error quantification campaign are displayed in pixel plot format for conditions at 1 and 35 bar pressure.  
Figure 4: Online and tabulated chemistry solutions for the progress variable (based on Equation (6)) evolution of a stoichiometric \(n\)-heptane/air mixture at 10 bar and 750 K.  
For a more effective error visualization as well as to better highlight the exact areas where interpolation errors may arise, pixel contours have been conditionally formatted so that every case having an error below 0.5% is displayed in white. All cases presenting an error above 0.5% will have contours consistent with the RGB color bar instead. For the whole range of computed conditions, the solution retrieved from the table did not exceed a 6% discrepancy on a single point basis, while the overall average error is approximately 1.5%. As expected, interpolation errors are more visible in the Negative Temperature Coefficient (NTC) regimes, which is the most challenging area to parametrize when progress variable methods are concerned [24; 55]. While a tighter tabulation grid in temperature space could have delivered even lower errors, the authors considered this level of accuracy to be acceptable. This decision was based on the globally low errors but also on method usability considerations.  
A tighter table grid results in a larger file size on disk, which in turn affects the random-access memory (RAM) requirement for the engine simulations during run-time. Hence, the present configuration was considered to be a good trade-off.