Bound-Preserving Discontinuous Galerkin Methods with Modified Patankar Time Integrations for Chemical Reacting Flows

In this paper, we develop bound-preserving discontinuous Galerkin (DG) methods for chemical reactive flows. There are several difficulties in constructing suitable numerical schemes. First of all, the density and internal energy positive, mass fraction each species is between 0 1. Second, due to rapid reaction rate, system may contain stiff sources, strong-stability-preserving explicit Runge-Kutta method result limited time-step sizes. To obtain physically relevant approximations, apply technique DG methods. Though traditional positivity-preserving techniques can successfully yield positive density, energy, fractions, they not enforce upper bound 1 fractions. solve problem, need (i) make sure fluxes equations fractions consistent with that equation density; (ii) choose conservative time integrations, such summation preserved. With above two conditions, have 1, then, all For discretization, modified Runge-Kutta/multi-step Patankar methods, which flux while implicit source. Such handle sources relatively large steps, preserve positivity target variables, keep be Finally, it straightforward combine integrations. The requires approximations at cell interfaces, pre-selected point values variables. match degree freedom, use $$Q^k$$ polynomials on rectangular meshes problems space dimensions. evolve time, first read Gaussian points. Then, slope limiters applied solutions those points, preserved by leading updated averages. addition, another limiter get used flux. Numerical examples given demonstrate good performance proposed