Comparison of different methods for source terms in detonation simulation
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摘要: 为解决爆震燃烧模拟中出现的刚性问题，对处理刚性源项问题常见的一步法、逼近法、拟稳态逼近法(α qusai steady state, αQSS)和点隐方法进行对比，从稳定性等方面分析源项处理方法应满足的时间步长要求，并探索各方法之间的联系以及适应化学反应特征变化的能力，进一步通过球头激波诱导燃烧算例比较每种方法的计算效率。理论分析和数值计算表明：一步法在积分刚性源项时，积分步长需小于或等于2倍最短反应特征时间，而逼近法、αQSS法和点隐方法对时间步长取值没有影响；αQSS法可根据化学反应特征的变化自动调整α值和时间步长，适用范围较广，而一步法和逼近法则是αQSS方法的特例。点隐等隐式方法在求解数学意义上的刚性问题时稳定性很好，但计算效率较低。相比而言，αQSS法在计算稳定性和适应化学反应变化方面都具有良好的性能，且针对激波诱导燃烧算例，αQSS法消耗的CPU时间仅为点隐方法的一半，是处理刚性源项较好的选择。Abstract: In this study, to solve the stiff source terms resulting from chemical reactions in detonation simulation, we examined the one step method, the asymptotic approach, the α quasi steady state method (αQSS) and the point implicit and compared their performances in coping with the stiff source term problems. We studied the limitations of each method using stability analysis, and investigated their relationships and capabilities in adapting to the changes in chemical reactions, with the shock-induced combustion simulated to compare their efficiencies. The results indicate that the one step method requires at least two times of the smallest time scale while the other three methods have no constraint on the time step. The αQSS can adjust the value of α and the time step with different reaction characteristics, and the one step method and the asymptotic method are the special cases of the αQSS with a constant α. An implicit approach has a better performance in mathematically solving the stiff equations but its low computation efficiency from the inversion of the matrix is sometimes unacceptable. The αQSS method can only consume a half of the CPU time that with the point implicit in shock-induced combustion simulation. In general, the αQSS is a good choice for dealing with stiff source term problems.
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Key words:
- stiff source /
- stability analysis /
- α quasi steady state /
- point implicit method /
- asymptotic method /
- one step method
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图 2 一步法处理组分质量分数随轴线分布曲线
Figure 2. Mass fraction distribution of species along the axis by one step method
3(a) αQSS方法处理组分质量分数随轴线分布曲线
3(a). Mass fraction distribution of species along the axis by αQSS method
3(b) 点隐法处理组分质量分数随轴线分布曲线
3(b). Mass fraction distribution of species along the axis by point implicit method
3(c) 逼近法处理组分质量分数随轴线分布曲线
3(c). Mass fraction distribution of species along the axis by asymptotic method
表 1 实验数据与理论模型结果对比
Table 1. Comparison of experimental and theoretical data
方法 f/Hz Ma=4.48 Ma=4.79 实验 425.0 712.0 αQSS 424.0 721.5 点隐法 426.1 725.7 一步法 424.1 721.5 逼近法 425.2 721.5 
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表 2 不同源项处理方法消耗CPU时间
Table 2. The CPU time in different stiff source term methods
方法 CPU time/s αQSS 77 590 点隐法 130 007 一步法 154 774 逼近法 62 372
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