Explicit time-scale splitting algorithm for stiff problems: auto-ignition of gaseous mixtures behind a steady shock
Explicit time-scale splitting algorithm for stiff problems: auto-ignition of gaseous mixtures behind a steady shock
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DOI：
10.1006/jcph.2001.6709
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发表时间：
2001-05
影响因子：
4.1
通讯作者：
M. Valorani;D. A. Goussis
中科院分区：
文献类型：
--
作者：
M. Valorani;D. A. Goussis
提出了一种新的基于计算奇异摄动（CSP）方法的显式算法。该算法是专门为求解刚性问题而设计的，其性能随刚度的增加而提高。其结构的关键概念是将问题中的快时间尺度从慢时间尺度中分离出来，通过将CSP概念嵌入到显式方案中实现。简而言之，该算法仅使用产生慢时间尺度的项进行时间行军，而在每个积分步骤结束时考虑产生快速时间尺度的项的贡献作为校正。针对偏微分方程的刚性系统，采用显式格式设计了新的积分算法。为了简单地介绍和讨论新算法的不同特征，考虑了一个简单的测试用例，涉及在正常冲击波后面的甲烷/空气混合物的自燃，这是由一个ode系统描述的。然后将新算法的性能（精度和计算效率）与众所周知的LSODE包进行比较。讨论了该方法在求解偏微分方程系统中的优点。虽然在处理刚性的偏微分方程系统时，新算法可以提供与LSODE相同的精度，但代价是更高的执行时间，但结果表明，在面对刚性的偏微分方程系统时，新算法的性能可能会更好。
A new explicit algorithm based on the computational singular perturbation (CSP) method is presented. This algorithm is specifically designed to solve stiff problems, and its performance increases with stiffness. The key concept in its structure is the splitting of the fast from the slow time scales in the problem, realized by embedding CSP concepts into an explicit scheme. In simple terms, the algorithm marches in time with only the terms producing the slow time scales, while the contribution of the terms producing the fast time scales is taken into account at the end of each integration step as a correction. The new algorithm is designed for the integration of stiff systems of PDEs by means of explicit schemes. For simplicity in the presentation and discussion of the different features of the new algorithm, a simple test case is considered, involving the auto-ignition of a methane/air mixture behind a normal shock wave, which is described by a system of ODEs. The performance of the new algorithm (accuracy and computational efficiency) is then compared with the well-known LSODE package. Its merits when used for the solution of systems of PDEs are discussed. Although when dealing with a stiff system of ODEs the new algorithm is shown to provide equal accuracy with that delivered by LSODE at the cost of higher execution time, the results indicate that its performance could be superior when facing a stiff system of PDEs.