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In this paper, we develop bound-preserving discontinuous Galerkin (DG)methods for chemical reactive flows. There are several difficulties inconstructing suitable numerical schemes. First of all, the density and internalenergy are positive, and the mass fraction of each species is between 0 and 1.Secondly, due to the rapid reaction rate, the system may contain stiff sources,and the strong-stability-preserving explicit Runge-Kutta method may result inlimited time step sizes. To obtain physically relevant numericalapproximations, we apply the bound-preserving technique to the DG methods. Fortime discretization, we apply the modified Runge-Kutta/multi-step Patankarmethods, which are explicit for the flux while implicit for the source. Suchmethods can handle stiff sources with relatively large time steps, preserve thepositivity of the target variables, and keep the summation of the massfractions to be 1. Finally, it is not straightforward to combine thebound-preserving DG methods and the Patankar time integrations. Thepositivity-preserving technique for DG method requires positive numericalapproximations at the cell interfaces, while Patankar methods can keep thepositivity of the pre-selected point-values of the target variables. To matchthe degree of freedom, we use $Q^k$ polynomials on rectangular meshes forproblems in two space dimensions. To evolve in time, we first read thepolynomials at the Gaussian points. Then suitable slope limiters can be appliedto enforce the positivity of the solutions at those points, which can bepreserved by the Patankar methods, leading to positive updated numerical cellaverages. In addition, we use another slope limiter to get positive solutionsused for the bound-preserving technique for the flux.

Bound-preserving discontinuous Galerkin methods with modified Patankar time integrations for chemical reacting flows