Space-time adaptive ADER-DG finite element method with LST-DG predictor and a posteriori sub-cell WENO finite-volume limiting for simulation of non-stationary compressible multicomponent reactive flows

TL;DR

The present work is devoted to the study of efficient implementation of spacetime adaptive ADER finite element discontinuous Galerkin method with a posteriori correction technique of solutions on subcells by the finite-volume ADER-WENO limiter scheme for simulation of non-stationary compressible multicomponent reactive flows.

Abstract

The space-time adaptive ADER finite element DG method with a posteriori correction technique of solutions on subcells by the finite-volume ADER-WENO limiter was used to simulate non-stationary compressible multicomponent reactive flows. The multicomponent composition of the reacting medium and the reactions occurring in it were described by expanding the original system of Euler equations by a system of non-stationary convection-reaction equations. The use of this method to simulate high stiff problems associated with reactions occurring in a multicomponent medium requires the use of the adaptive change in the time step. The solution of the classical problem related to the formation and propagation of a ZND detonation wave is carried out. It was shown that the space-time adaptive ADER finite element DG method with a posteriori correction technique of solutions on subcells by the finite-volume ADER-WENO limiter can be used to simulate flows without using of splitting in directions and fractional step methods.

Figures (11)

• Figure 1: Numerical solution of the classical problem of advection flow for a multicomponent medium (a detailed statement of the problem is presented in the text), using the computational scheme $\mathrm{ADER}$-$\mathrm{DG}$-$\mathbb{P}_5$ with a posteriori limitation of the solution by a $\mathrm{ADER}$-$\mathrm{WENO}5$ finite volume limiter, on a coordinate mesh with $5$ (top figures) and $800$ (bottom figures) finite element cells. Each finite element contains $N_{s} = 11$ subcells. In the top figures, vertical lines indicate the coordinate boundaries of each finite element. The graphs show the coordinate dependencies of subcells finite-volume representation of density $\rho$ (a, d) and densities $\rho_{k} = \rho c_{k}$ (b, c, e, f) of individual components $k = 1,\, 2$ of the multicomponent medium, at the final time $t_{\rm final} = 1.0$. The black square symbols represent the numerical solution; the red circle symbols represents the exact analytical solution of the problem.
• Figure 2: Numerical solution of the classical Sod problem for a multicomponent medium (a detailed statement of the problem is presented in the text), using the computational scheme $\mathrm{ADER}$-$\mathbb{P}_5$ with a posteriori limitation of the solution by a $\mathrm{ADER}$-$\mathrm{WENO}5$ finite volume limiter, on a coordinate mesh with $1800$ finite element cells. The graphs show the coordinate dependencies of pressure $p$ (a), density $\rho$ (b), flow velocity $u$ (c), sound speed $c$ (d), and densities $\rho_{k} = \rho c_{k}$ (e1-e4) of individual components $k$ of the multicomponent medium, at the final time $t_{\rm final} = 0.15$. The black square symbols represent the numerical solution; the red solid lines represents the exact analytical solution of the problem.
• Figure 3: Numerical solution of the classical Lax problem for a multicomponent medium (a detailed statement of the problem is presented in the text), using the computational scheme $\mathrm{ADER}$-$\mathrm{DG}$-$\mathbb{P}_4$ with a posteriori limitation of the solution by a $\mathrm{ADER}$-$\mathrm{WENO}4$ finite volume limiter, on a coordinate mesh with $1800$ finite element cells. The graphs show the coordinate dependencies of pressure $p$ (a), density $\rho$ (b), flow velocity $u$ (c), sound speed $c$ (d), and densities $\rho_{k} = \rho c_{k}$ (e1-e4) of individual components $k$ of the multicomponent medium, at the final time $t_{\rm final} = 0.15$. The black square symbols represent the numerical solution; the red solid lines represents the exact analytical solution of the problem.
• Figure 4: Numerical solution of the classical problem with two rarefaction waves for a multicomponent medium (a detailed statement of the problem is presented in the text), using the computational scheme $\mathrm{ADER}$-$\mathrm{DG}$-$\mathbb{P}_5$ with a posteriori limitation of the solution by a $\mathrm{ADER}$-$\mathrm{WENO}5$ finite volume limiter, on a coordinate mesh with $1800$ finite element cells. The graphs show the coordinate dependencies of pressure $p$ (a), density $\rho$ (b), flow velocity $u$ (c), sound speed $c$ (d), and densities $\rho_{k} = \rho c_{k}$ (e1-e4) of individual components $k$ of the multicomponent medium, at the final time $t_{\rm final} = 0.15$. The black square symbols represent the numerical solution; the red solid lines represents the exact analytical solution of the problem.
• Figure 5: Numerical solution of the classical problem with two shock waves for a multicomponent medium (a detailed statement of the problem is presented in the text), using the computational scheme $\mathrm{ADER}$-$\mathrm{DG}$-$\mathbb{P}_5$ with a posteriori limitation of the solution by a $\mathrm{ADER}$-$\mathrm{WENO}5$ finite volume limiter, on a coordinate mesh with $1800$ finite element cells. The graphs show the coordinate dependencies of pressure $p$ (a), density $\rho$ (b), flow velocity $u$ (c), sound speed $c$ (d), and densities $\rho_{k} = \rho c_{k}$ (e1-e4) of individual components $k$ of the multicomponent medium, at the final time $t_{\rm final} = 0.15$. The black square symbols represent the numerical solution; the red solid lines represents the exact analytical solution of the problem.