# Application of the Projective Method in the Numerical Simulation of Combustion

## Abstract

The numerical simulation of combustion is one of the most important research tools for developing alternative fuels and high-performance combustion engines, which is widely used in the field of aerospace. However, the broad range of time scales and complex chemical kinetics bring great challenge to combustion simulation. With traditional implicit methods, most of the CPU time will be spent on solving the stiff ODEs caused by detailed chemistry mechanism. So, the object of this study is to use an accurate and efficient explicit method to solve the stiff ODEs, by which the computational efficiency can be greatly improved. Developed by Gear and co-workers, the projective method is utilized to solve stiff ODEs with a broad range of time scales, which is suitable for combustion problems. In this study, the homogeneous ignition process and spherical flame propagation process of methane are both investigated. Compared to the results from a traditional implicit method, VODE, projective method gives almost the same results, while the calculation speed is one-order faster than that of the VODE method. Due to its high accuracy and efficiency, the projective method will have great potentials in combustion simulation.

## Keywords

Projective method Combustion Numerical simulation Stiff equations## 1 Introduction

Combustion and its relevant theories are widely used in propulsion technology in the field of aerospace. As a powerful tool, numerical simulation is helpful for understanding fundamental combustion phenomena and for revealing the physical-chemical mechanisms in combustion processes. Therefore, numerical simulation becomes one of the most important research tools for developing alternative fuels and high-performance combustion engines [4].

However, combustion involves multiple physical-chemical processes, broad range of time and spatial scales and complex chemical kinetics, bringing great challenge to numerical simulation [14]. For accurate simulation and quantitative prediction, detailed chemistry must be investigated, which will cause significant difficulty to simulation. On one hand, different components have different characteristic time, leading to a wide range of time scale. For example, the characteristic time of hydrogen and methane is 10^{−9} to 10^{−5} s, while for n-heptane, it is 10^{−13} to 10^{−5} s [1]. So, with the increase of complexity of molecule structure, the time scale of components in detailed chemistry will be more and more extensive, leading to very severe stiffness of relevant differential equations. One the other hand, with the increase of carbon number, the number of components and elementary reactions will increase exponentially, which also leads to a sharp increase of computation quantity [9]. For example, for fuels such as n-heptane and iso-octane, the number of components in detailed chemistry is about one thousand. As a result, nearly one thousand conservation equations concerning each component need to be solved, leading to a huge challenge to the simulation. During the simulation of propagation of methane/air premixed flame, solving stiff ODEs caused by detailed chemistry mechanisms will take most of the CPU time, if the traditional implicit method VODE is used. So, the key point to simulate the chemistry reaction flow efficiently is to deal with the wide range of time scales and complex chemistry kinetics mechanism.

To solve the stiff problem caused by the diverse time scale of different components, researchers proposed kinds of methods. For example, quasi-steady-state approximation (QSSA) and partial equilibrium approximation (PEA) [13] are classical methods to reduce stiffness. However, both of them need to experiential prediction and are difficult to adapt the constantly-changing conditions in the process of combustion. Recently, Lu and co-workers put forward a correctional method [10], which employed a different way to deal with QSS and PE components, according to the sparseness of complex coupling, to reduce the stiffness. But a shortage of this method is the difficulty to use explicit integration with big time step. Another way to solve stiff problems is to develop accurate numerical methods with high efficiency to solve stiff ODEs. Traditionally, implicit methods, such as VODE, are often employed to solve stiff ODEs, in which Jacobi Matrices are needed to be calculated. Thus the computation quantity is proportional to the square or even cube of the component number, which influence the computation efficiency very much. So, there are researchers who try to advance the explicit method or develop explicit-implicit-multi methods. For instance, Gou and co-workers recently develop a multi-time-scale method (MTS) [7, 8], by which the components are divided into different groups according to their characteristic time. In the fast group and slow group, different time step is used to conduct the Euler integration. As a result, the computation speed can be enhanced by one order. However, as the mass conservation of components cannot be ensured during the computation process, sometimes the error will accumulate to some extent and lead to a mistake.

In this paper, a totally explicit numerical algorithm, the projective method [5, 6], will be employed to simulate the homogeneous ignition process and spherical flame propagation process of methane, with the computation result and efficiency compared to that of implicit VODE method and MTS method, to reveal the advantage of the projective method in dealing with complex chemistry mechanism.

## 2 Principles of the Projective Method

*y*is a vector function of

*m*dimension. It is supposed that

*λ*

_{j}is eigenvalue of Matrix A and all of them are negative. The stiff ODEs mean that there are faster-change and slower-change parts in the solution, corresponding to bigger and smaller eigenvalues respectively. Faster-change parts only contribute to the solution in a very short time, and then the slower-change parts dominate the solution. Nevertheless, the stability requirement of numerical method is required by faster-change parts all the time. So, to get the numerical integration solution, time step is strictly constrained by the faster-change parts but the length of integration interval is decided by slower-change parts, leading to a huge number of integration steps, which is the main difficulty for solving the numerical solution of stiff ODEs.

*Projective Method*(referred to as ‘PM’ in the following text) firstly in 2003 to solve stiff ODEs with a broad range of time scales. The basic principle of PM is as follows: Initially, integrate for some steps with a small time step, corresponding to time constant of faster-change parts, to decrease the disturbed error and faster-change parts of the solution, which could be called ‘inner layer integration’. Next, the character of the solution is mainly depended on the slower-change parts, so the trend could be simulated on a longer interval by an extrapolation based on the last points of inner integration. For example, if the last two points of inner integration is used to conduct a linear extrapolation, the trend of slower-change parts is approximately represented by the slope of the line connected by that two points, which could be called Projective Forward Euler method [5]. Suppose the length of inner time step is

*h*and numerical solution

*y*

_{n}at time

*t*=

*t*

_{n}has been obtained, then the calculate procedure in next (

*k*+

*1*+

*M*) time steps is listed as follows:

- (1)
from

*t*_{n}to*t*_{n + k}, integrate with explicit method (e.g. Euler method) for*k*steps. Here*h*is rather smaller, so the high order accuracy is not necessary and basic requirement is just stability; - (2)
from

*t*_{n + k}to*t*_{n + k + 1}, conduct inner integration for another step, and*y*_{n + k + 1}will be obtained; - (3)
base on

*y*_{n + k}and*y*_{n + k + 1}, extrapolate to an interval with length equaling to*Mh*, that is,*y*_{n + k + 1 + M}=*(M*+*1) y*_{n + k + 1}−*My*_{n + k}

*k*+

*1*) steps and an outer extrapolation on an interval of

*Mh*, which could be called ‘P

*kM*’ method. Figure 1 gives a sketch of the projective method [5] with k = 2.

According to the opinion of Gear etc., the procedure above can be ‘iterated’ [6], that is to say, a ‘P*kM*’ procedure can be packaged as a whole step with length equaling to *h*_{1} = (*k* + *1* + *M*)*h*, which can be regarded a new inner layer. Based on the new layer, several step of integration and an extrapolation on *Mh*_{1} interval can be conducted, where a longer step *h*_{2} = (*k* + *1* + *M*)*h*_{1} can be obtained. By this kind of procedure, multiple layers of integration step can be built, which is named ‘Telescopic Projective Method’ by Gear etc. [6]. Usually, a two-layer projective method could be named as P*k*_{1}*M*_{1}−*k*_{2}*M*_{2.}

## 3 Simulation for Homogeneous Ignition Process by Projective Method

In this section, a zero-dimension insulated homogeneous system with stable mass and pressure is studied, with pressure set to 1 atm and equivalent ration set to 1.0.

For projective method (PM), the integration step of innermost layer is set to *h *= 10^{−9}s, which is close to the maximum step that satisfies the stability of explicit Euler method. Based on this, PM of one, two or three layers could be built, and corresponding outer step is 10^{−8}s, 10^{−7}s and 10^{−6}s. The result of PM is compared to that of VODE and, to evaluate the calculation efficiency of different methods, the ratio of the CPU time cost of VODE against other methods with same *t*_{base} (for PM, *t*_{base} is the outer layer step) is defined as speed-up. The speed-up of VODE is defined as 1.

_{4}and OH, with initial temperate

*T*

_{0}= 1400 K. Figures 2(a) and (b) are results of one and two layer PM compared with VODE. It is demonstrated that the calculation results from PM and VODE are almost the same.

## 4 Simulation for Flame Propagation Process by Projective Method

In the simulation of homogeneous ignition process discussed above, only chemistry reaction is considered. In this section, projective method will be applied in the simulation of flame propagation process, which contains both chemistry reaction and transport process like convection and diffusion, to validate the accuracy and efficiency of projective method.

An adaptive simulation program of unsteady reaction flow (A-SURF) is applied in this section to simulate the propagation of one dimension sphere flame of methane/air. A-SURF is widely applied in the simulation of sphere flame [2, 3], the governing equations of that are conservation equations of 1-D unsteady reaction flow in spherical coordinate system. More detailed information about A-SURF could be referred to the reference [3], and it is worth mentioning that a ‘splitting’ strategy is adopted in the program to separate the calculation process into two steps [12]: one is solving the partial differential equations (PDEs) only for convection and diffusion, the other is solving ODEs only describing the chemistry reaction, which is in fact the homogeneous ignition process discussed in last section.

_{4}and CO at time t = 3 ms. It is demonstrated that results obtained by PM and VODE are almost the same, which validates that PM can simulate the process of flame propagation accurately.

Time cost for simulation of spherical flame propagation until t = 1 ms

CH | P18 | VODE | Speed-up | ||
---|---|---|---|---|---|

Time(s) | Proportion | Time(s) | Proportion | ||

Total | 12400 | 100% | 192760 | 100% | 15.55 |

PDEs | 6230.4 | 50.2% | 6313.2 | 3.3% | 1.01 |

ODEs | 6132.8 | 49.5% | 186409 | 96.7% | 30.40 |

Diffusion | 3909.2 | 31.5% | 3972.6 | 2.1% | 1.02 |

Viscosity | 1435.6 | 11.6% | 1428.9 | 0.7% | 1.00 |

Others | 885.6 | 7.1% | 911.7 | 0.5% | 1.03 |

## 5 Conclusions

- (1)
The projective method can guarantee the calculation accuracy. For methane ignition and flame propagation, it accurately predicts the change of temperature and main components over time and space, and the process of flame propagation.

- (2)
The projective method can greatly improve computational efficiency. Compared with the conventional VODE method, the projective method can enhance the computational speed by approximately one order of magnitude. Due to this, the projective method will have great potentials in combustion simulation.

## References

- 1.Chen Z (2009) Multi-scale Simulation of Chemistry Reaction Flow. In: CCTAM2009, Zhengzhou, ChinaGoogle Scholar
- 2.Chen Z, Burke MP, Ju Y (2009) Effects of compression and stretch on the determination of laminar flame speeds using propagating spherical flames. Combust Theor Model 13:343–364CrossRefGoogle Scholar
- 3.Chen Z, Burke MP, Ju Y (2009) Effects of Lewis number and ignition energy on the determination of laminar flame speed using propagating spherical flames. Proc Combust Inst 32:1253–1260CrossRefGoogle Scholar
- 4.Dec J (2009) Advanced compression-ignition engines-understanding the in-cylinder processes. Proc Combust Inst 32:2727–2742CrossRefGoogle Scholar
- 5.Gear C, Kevrekidis IG (2003) Projective methods for stiff differential equations: problems with gaps in their eigenvalue spectrum. SIAM J Sci Comput 24(4):1091–1106MathSciNetCrossRefGoogle Scholar
- 6.Gear C, Kevrekidis IG (2003) Telescopic projective methods for parabolic differential equations. J Comput Phys 187:95–109MathSciNetCrossRefGoogle Scholar
- 7.Gou X, Sun W, Chen Z, Ju Y (2010) A dynamic multi-timescale method for combustion modeling with detailed and reduced chemical kinetic mechanisms. Combust Flame 157(6):1111–1121CrossRefGoogle Scholar
- 8.Gou X, Sun W, Chen Z, Ju Y (2010) Multi-timescales simulation for air ignition process. Science and technology of Combustion 16(5):452–455Google Scholar
- 9.Lu T, Law C (2009) Towards accommodating realistic fuel chemistry in large scale computations. Prog Energy Combust Sci 35:192–215CrossRefGoogle Scholar
- 10.Lu T, Law C, Yoo C, Chen J (2009) Dynamic stiffness removal for direct numerical simulations. Combust Flame 156:1542–1551CrossRefGoogle Scholar
- 11.Smith G et al GRI-MECH 3.0. http://www.me.berkeley.edu/grimech
- 12.Strang G (1968) On the construction and comparison of difference schemes. SIAM J Numer Anal 5:506–517MathSciNetCrossRefGoogle Scholar
- 13.Turns SR (2009) An Introduction to Combustion: Concepts and Applications, 2nd ednGoogle Scholar
- 14.Westbrook C, Mizobuchi Y, Poinsot T, Smith P, Warnatz E (2005) Computational combustion. Proc Combust Inst 30:125–157CrossRefGoogle Scholar