Accurate deterministic projection methods for stiff detonation waves

We study numerical approximations of the reactive Euler equations of gas dynamics. In addition to shock, contact and rarefaction waves, these equations admit detonation waves appearing at the interface between different fractions of the reacting species. It is well-known that in order to resolve the reaction zone numerically, one has to take both space and time stepsizes to be proportional to the reaction time, which may cause the numerical method to become very computationally expensive or even impractical when the reaction is fast. Therefore, it is necessary to develop underresolved numerical methods, which are capable of accurately predicting locations of the detonation waves without resolving their detailed structure. One can distinguish between two different degrees of stiffness. In the stiff case, the reaction time is very small, while in the extremely stiff case, the reaction is assumed to occur instantaneously.

In [A. Kurganov, $\href{https://link.springer.com/book/10.1007/978-3-540-75712-2}{\textit{Hyperbolic problems: theory, numerics, applications}}$, Springer, Berlin, 2003], we proposed a simple underresolved method—an accurate deterministic projection (ADP) method—for one-dimensional hyperbolic systems with stiff source terms including the reactive Euler equations in the extremely stiff regime. In this paper, we extend the ADP method to the (non-extremely) stiff case, multispecies detonation models, and the two-dimensional reactive Euler equations in all of the aforementioned regimes. We also investigate ways to distinguish between different regimes in practice as well as study the limitations of the proposed ADP methods with respect to the ignition temperature. We demonstrate the accuracy and robustness of the ADP methods in a number of numerical experiments with both relatively low and large ignition temperature, and illustrate the difficulties one may face when the ignition temperature is low.

Keywords

stiff detonation waves, reactive Euler equations, splitting method, deterministic projection method, central-upwind scheme, multispecies detonation

2010 Mathematics Subject Classification

35L65, 35L67, 65M08, 76M12, 76V05

Full Text (PDF format)

Received 21 February 2022

Received revised 11 August 2023

Accepted 7 September 2023

Published 12 July 2024