Higher Order Accurate Bounds Preserving Time-Implicit Discretizations for the Chemically Reactive Euler Equations

Abstract

We construct higher order accurate bounds preserving time-implicit Discontinuous Galerkin (DG) discretizations for the reactive Euler equations modelling multispecies and multireaction chemically reactive flows. In numerical discretizations of chemically reactive flows, the time step can be significantly limited because of the large difference between the fluid dynamics time scales and the reaction time scales. In addition, the density and pressure should be nonnegative and the mass fractions between zero and one, which imposes constraints on the numerical solution that must be satisfied to obtain physically reliable solutions. We address these issues using the following steps. Firstly, we develop the Karush-Kuhn-Tucker (KKT) limiter for the chemically reactive Euler equations, which imposes bounds on the numerical solution using Lagrange multipliers, and solve the resulting KKT mixed complementarity problem using a semi-smooth Newton method. The disparity in time scales is addressed using a fractional step method, separating the convection and reaction steps, and the use of higher order accurate Diagonally Implicit Runge-Kutta (DIRK) methods. Finally, Harten’s subcell resolution technique is used to deal with stiff source terms in chemically reactive flows. Numerical results are shown to demonstrate that the bounds preserving KKT-DIRK-DG discretizations are higher order accurate for smooth solutions and able to capture complicated stiff multispecies and multireaction flows with discontinuities.