High-Order Bound-Preserving Discontinuous Galerkin Methods for Stiff Multispecies Detonation

Abstract

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 z i , 1 ≤ i ≤ M , should be between 0 and 1, where M is the total number of species. In [18], the authors have introduced the positivity-preserving technique that guarantee 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 [18] 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 highorder 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 [18], 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.

A hybrid lattice Boltzmann method for gaseous detonations

“…The intricate interplay of shocks, chemical reactions, and turbulence therefore requires the use of very robust and low dissipative numerical methods [6]. To tackle these challenges, various strategies have been embraced in the literature, ranging from the use of high-order finite volume schemes based on HLLC (Harten-Lax-van Leer-Contact) solvers [7][8][9][10] to Galerkin discontinuous methods [11,12]. Despite significant enhancements in modeling strategies through adaptive mesh refinement [13][14][15], simulating an entire detonation engine remains computationally challenging with these high-order approaches.…”

Positivity-preserving and entropy-bounded discontinuous Galerkin method for the chemically reacting, compressible Navier-Stokes equations

“…Entropy boundedness was not considered. Instead of operator splitting, they employed an exponential multistage/multistep, explicit time integration scheme [18,19,20] that can handle stiff source terms. Although in the present study we use operator splitting since it has proven successful to date and its accuracy is less reliant on "well-prepared" initial conditions [18,19,20], exponential multistage/multistep time integrators are indeed worthy of future investigation.…”

Is the Classic Convex Decomposition Optimal for Bound-Preserving Schemes in Multiple Dimensions?

“…In the pioneering work of [1,2], Zhang and Shu proposed a general framework of designing high-order BP discontinuous Galerkin (DG) and finite volume (FV) schemes for hyperbolic conservation laws on rectangular meshes, later generalized to triangular meshes in [3]. Over the past decade, the Zhang-Shu framework has attracted extensive attention and been applied to various hyperbolic systems (e.g., [4,5,6,7,8,9,10,11,12]) and convection-dominated equations (e.g., [13,14,15,16,17]). Recently, motivated by a series of BP works [18,19,20,21] for magnetohydrodynamics, the geometric quasilinearization (GQL) framework was proposed in [22] for studying BP problems involving nonlinear constraints.…”