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We propose an operator-splitting scheme to approximate scalar conservation equations with stiff source terms having multiple (at least two) stable equilibrium points. The scheme com- bines a (reaction-free) transport substep followed by a (transport-free) reaction substep. The transport substep is approximated using the forward Euler method with continuous finite elements and graph viscosity. The reaction substep is approximated using an exponential integrator. The crucial idea of the paper is to use a mesh-dependent cutoff of the reaction time-scale in the reaction substep. We establish a bound on the entropy residual motivating the design of the scheme. We show that the proposed scheme is invariant-domain preserv- ing under the same CFL restriction on the time step as in the nonreactive case. Numerical experiments in one and two space dimensions using linear, convex, and nonconvex fluxes with smooth and nonsmooth initial data in various regimes show that the proposed scheme is asymptotic preserving.
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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, in 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.
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- PAR ID:
- 10535731
- Publisher / Repository:
- International Press
- Date Published:
- Journal Name:
- Communications in Mathematical Sciences
- Volume:
- 22
- Issue:
- 4
- ISSN:
- 1539-6746
- Page Range / eLocation ID:
- 871 to 910
- Subject(s) / Keyword(s):
- Stiff detonation waves reactive Euler equations splitting method deterministic projection method central-upwind scheme multispecies detonation.
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.4310/CMS.2024.v22.n4.a1
