Application of the Projective Method in the Numerical Simulation of Combustion

  • Yang LiuEmail author
  • Zheng Chen
Conference paper
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 459)


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.


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

Considering the following linear ordinary differential equations (ODEs):
$$ {{dy} \mathord{\left/ {\vphantom {{dy} {dt}}} \right. \kern-0pt} {dt}} = Ay + g(t) $$
where y is a vector function of m dimension. It is supposed that λj 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.
Gear and co-workers put forward the so-called 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 yn at time t = tn has been obtained, then the calculate procedure in next (k + 1 + M) time steps is listed as follows:
  1. (1)

    from tn to tn + 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. (2)

    from tn + k to tn + k + 1, conduct inner integration for another step, and yn + k + 1 will be obtained;

  3. (3)

    base on yn + k and yn + k + 1, extrapolate to an interval with length equaling to Mh, that is, yn + k + 1 + M = (M + 1) yn + k + 1− Myn + k

The key point of above procedure is inner integration for (k + 1) steps and an outer extrapolation on an interval of Mh, which could be called ‘PkM’ method. Figure 1 gives a sketch of the projective method [5] with k = 2.
Fig. 1.

Procedure of P2M method

According to the opinion of Gear etc., the procedure above can be ‘iterated’ [6], that is to say, a ‘PkM’ procedure can be packaged as a whole step with length equaling to h1 = (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 Mh1 interval can be conducted, where a longer step h2 = (k + 1 + M)h1 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 Pk1M1k2M2.

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 = 10−9s, 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 10−8s, 10−7s and 10−6s. 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 tbase (for PM, tbase is the outer layer step) is defined as speed-up. The speed-up of VODE is defined as 1.

For methane/air ignition process, a detailed chemistry mechanism called GRI-Mech3.0 [11] is employed, which contains 53 kinds of components and 325 base reactions. Figure 2 illustrates the time history of temperature and mass percentage of CH4 and OH, with initial temperate T0 = 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.
Fig. 2.

Time history of temperature and mass percentage of CH4 and OH

As to the calculation efficiency, Fig. 3 show the speed-up of different methods. It reveals that the efficiency of PM rises with the decrease of number of layer. For one-layer PM, the calculation speed is 30 times faster than VODE. So, PM can be used to simulate the process of methane homogeneous ignition with good accuracy and high efficiency.
Fig. 3.

Speed-up of different methods

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.

Figure 4 shows the calculation results of the methane flame propagation, where Fig. 4(a) is the time history of flame radius and Fig. 4(b) is the spatial distribution of temperature, heat release rate and mass percentage of CH4 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.
Fig. 4.

Results of spherical flame propagation of CH4/air

Table 1 gives the CPU time cost of different methods. The physical time is from 0 to 1 ms while the length of time step is 10 ns, so there are totally 100,000 time steps. The ‘speed-up’ in the table is the ratio of the CPU time cost of VODE against PM, and this definition is the same with that in Sect. 2.
Table 1.

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













































Because of the splitting strategy, the two methods only have difference in the solution of stiff ODEs, so the computation time on other terms is theoretically the same, that is, the speed-up is one. A little error (within 3%) in Table 1 is caused by CPU disturbance during the calculation. Table 1 illustrates that if VODE method is employed, most CPU time is occupied by solving stiff ODEs. So the key point of enhancing the efficiency of combustion simulation is how to solve the stiff ODEs caused by detailed chemistry mechanism rapidly. Table 1 shows that if PM is employed, the time cost of ODEs is only 1/30 of that of VODE, and the total time cost is 1/15 of VODE. So, as to the simulation of flame propagation, PM can enhance the calculation speed by one order. Figure 5 illustrates the time cost percentage of each term in the equations, which obviously shows that time cost of solving stiff ODEs decreases significantly with PM replacing VODE method.
Fig. 5.

Time cost percentage for solving each item of equations

5 Conclusions

In this paper, the projective method is applied to the numerical simulation of methane combustion process. The research object includes both the zero-dimensional homogeneous ignition problem involving chemical reaction only, and the flame propagation problem involving both the chemical reaction and transport (convection and diffusion) process. Numerical test results show that:
  1. (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. (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.



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© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Shanghai Aircraft Design and Research InstituteShanghaiChina
  2. 2.College of EngineeringPeking UniversityBeijingChina

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