Are the “stiff systems” of differential equations so stiff?
Abstract
Using the example of a well-known stiff Robertson reaction scheme, the practical application of various methods for integrating the direct problem of chemical kinetics is shown. A comparison of one-step and multi-step methods, explicit and implicit solution schemes was made and an original approach was developed that combines the application of the Adams method at the initial stage of reaction development with the transition to a system of differential-algebraic equations solved by the Euler method after achieving a stationary mode. A record short total computer time has been reached for the numerical solution of the problem over a time interval of more than 20 orders of magnitude. The proposed method is at least one hundred times faster than traditional algorithms.
References
Vasilyev, E. I., Vasilyeva, T. A., & Kiseleva M.N. (2013). L-stability of multi-implicit methods of 8-th order for differential stiff systems. Mathematical Physics and Computer Modeling. 1(18), 70–83. (in Russ.).
Zausaev A. F. (2010). Difference methods for solving ordinary differential equations. Samara: Samara State Technical University. (in Russ.).
Chen, R. T., Rubanova, Y., Bettencourt, J., & Duvenaud, D. K. (2018). Neural ordinary differential equations. Advances in neural information processing systems, 31.
Hairer, E., Wanner, G., & Nørsett, S. P. (1993). Solving ordinary differential equations I: Nonstiff problems. Berlin, Heidelberg: Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-540-78862-1.
Hairer, E., & Wanner, G. (1991). II: Stiff and differential-algebraic problems. Berlin [etc.]: Springer. https://doi.org/10.1007/978-3-662-09947-6.
Novikov E. A. Explicit Methods for Stiff Systems. Novosibirsk: Science SB RAS, 1997, 195 p. (in Russ.).
Robertson, H. H. (1966). The solution of a set of reaction rate equations. Numerical analysis: an introduction, 178182, 31. (in Russ.).
Novikov E. A., & Zakharov A .A. (2015). Variable structure algorithm with the rosenbrock method of a third-order approximation applied. Bulletin of Tyumen State University, Physical and Mathematical Modeling. Oil, Gas, Energy, 1(1), 146‒154. (in Russ.)
Gulevich, D. R., & Zalipaev, V.V. (2020). Numerical Methods in Physics and Engineering. St. Petersburg: UITMO, 211 p. (in Russ.).
Pinchukov, V. I. (2002). On a numerical solution of equations of a viscous gas by a third-order implicit Runge–Kutta scheme. Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, 42(6), 896‒904.
Kidger, P. (2022). On neural differential equations. arXiv preprint arXiv:2202.02435. (Doctoral thesis), Trinity: Mathematical Institute University of Oxford. https://doi.org/10.48550/arXiv.2202.02435.
Lagaris, I. E., Likas, A., & Fotiadis, D. I. (1998). Artificial neural networks for solving ordinary and partial differential equations. IEEE transactions on neural networks, 9(5), 987‒1000. https://doi.org/10.1109/72.712178.
Zhang, T., Zhang, Y., & Ju, Y. (2020). A deep learning-based ODE solver for chemical kinetics. arXiv preprint arXiv:2012.12654.
Ji, W., Qiu, W., Shi, Z., Pan, S., & Deng, S. (2021). Stiff-pinn: Physics-informed neural network for stiff chemical kinetics. The Journal of Physical Chemistry A, 125(36), 8098‒8106. https://doi.org/10.1021/acs.jpca.1c05102.
Gomes, C., Thule, C., Broman, D., Larsen, P. G., & Vangheluwe, H. (2018). Co-simulation: a survey. ACM Computing Surveys (CSUR), 51(3), 1‒33. https://doi.org/10.1145/3179993.
Copyright (c) 2025 Sergey O. Travin
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
