High-order bound-preserving discontinuous Galerkin methods for stiff multispecies detonation

In this paper, we develop high-order bound-preserving discontinuous Galerkin (DG) methods for multispecies and multireaction chemical reactive flows. In this problem, density and pressure are nonnegative, and the mass fraction for the ith species, denoted as zi, 1 ≤ i ≤ M, should be between 0 and 1, where M is the total number of species. In [C. Wang, X. Zhang, C.-W. Shu, and J. Ning, J. Comput. Phys., 231 (2012), pp. 653-665], the authors have introduced the positivity-preserving technique that guarantees the positivity of the numerical density, pressure, and the mass fraction of the first M − 1 species. However, the extension to preserve the upper bound 1 of the mass fraction is not straightforward. There are three main difficulties. First of all, the time discretization in [C. Wang, X. Zhang, C.-W. Shu, and J. Ning, J. Comput. Phys., 231 (2012), pp. 653-665] was based on Euler forward. Therefore, for problems with stiff source, the time step will be significantly limited. Secondly, the mass fraction does not satisfy a maximum principle, and most of the previous techniques cannot be applied. Thirdly, in most of the previous works for gaseous denotation, the algorithm relies on the second-order Strang splitting methods where the flux and stiff source terms can be solved separately, and the extension to high-order time discretization seems to be complicated. In this paper, we will solve all the three problems given above. The high-order time integration does not depend on the Strang splitting; i.e., we do not split the flux and the stiff source terms. Moreover, the time discretization is explicit and can handle the stiff source with large time step. Most importantly, in addition to the positivity-preserving property introduced in [C. Wang, X. Zhang, C.-W. Shu, and J. Ning, J. Comput. Phys., 231 (2012), pp. 653-665], the algorithm can preserve the upper bound 1 for each species. Numerical experiments will be given to demonstrate the good performance of the bound-preserving technique and the stability of the scheme for problems with stiff source terms.