## 2 Simulation Methods
### Combustion Progress Variable Model  
The CPV model relies on the pre-tabulation of the auto-ignition and emission formation processes for a broad range of initial conditions in terms of exhaust gas recirculation rates (yEGR), pressure (p), unburned temperature (T\({}_{\rm u}\)) and equivalence ratio (\(\phi\)). The reconstruction of the thermo-chemical state during run-time is then performed by means of an appropriately chosen progress variable (herein noted as C). As discussed in detail by [PERSON] et al. [51] and [PERSON] et al. [52], any progress variable-based method needs to fulfill the following conditions to be mathematically sound: (1) strict monotonicity with time; (2) non-negativity and normalization (C = 0 for unreacted mixture and C = 1 for fully burned mixture); (3) if reactive scalars (i.e., chemical species) are used in the definition of the progress variable, said scalars should evolve on comparable time scales [52]. Several progress variable definitions have been proposed by different groups for both homogeneous and flamelet-based tabulation frameworks [10; 13; 21]. The most common approach consists in formulating the progress variable via a combination of chemical species (typically including the surrogate fuel molecule, O\({}_{2}\), CO, CO\({}_{2}\), H\({}_{2}\)O, OH, CH\({}_{2}\)O and others). The choice of the species to use and their weighting factors within the progress variable definition is usually optimized so that both low and high temperature combustion regimes are captured [51; 52]. In the present work, latent enthalpy (enthalpy of formation at standard state, h\({}_{298}\)) is used to define the reaction progress variable C as reported below in a normalized fashion. \[\rm{C(t)}=\frac{h_{298}(t)-h_{298,u}}{h_{298,max}-h_{298,u}} \tag{6}\]  
In Equation (6), h\({}_{298}\) is the latent enthalpy calculated at 298 K, and summed over all species, subscript u denotes unburned state, and max denotes the most reacted state, which is assumed to be where the maximum of the accumulated chemical heat is released. As discussed in [15; 33], this progress variable can be used to track both low and high temperature reactions, which is important when tabulation methods are developed for fuels exhibiting cool flames. Figure 3 shows an exemplary h\({}_{298}\) profile together with temperature for a constant pressure reactor calculation. It can be discussed that in a constant pressure reactor, progress variables based on temperature (i.e., as proposed by [PERSON] et al. [24]) or on latent enthalpy will be equal. However, unlike temperature the latent enthalpy will not be influenced by pressure changes, fuel vaporization or mixing. As h\({}_{298}\)  
Figure 3: Temperature (black) and latent enthalpy at 298 K (red) as function of time for a constant pressure reactor calculation at 10 bar and 750 K for stoichiometric \(n\)-heptane/air mixture. is a conserved scalar, a transport equation, as well as a spray source term, can be easily formulated. For the majority of the tabulated conditions, the maximum of the accumulated chemical heat release coincides with the chemical equilibrium state. However, for fuel-rich conditions as well as for states that lead to pronounced endothermic reactions (i.e., high pressure or high EGR conditions), the mixture relaxation towards equilibrium is not taken into account by Equation (6). Nevertheless, this definition was considered acceptable for engine applications in the SRM since it affects a small fraction of the particles for a limited portion of the cycle in direct injected-engine simulations. Over the whole range of computed conditions, this definition was found to be convenient to correctly capture both low and high temperature combustion regimes when generating the table through adiabatic homogeneous constant pressure reactor calculations. To keep the table size within a feasible range for engine applications, storage of the progress variable source terms was done over 25 points for each reactor simulation.