Space-time adaptive ADER-DG finite element method with LST-DG predictor and a posteriori sub-cell ADER-WENO finite-volume limiting for multidimensional detonation waves simulation

TL;DR

The paper advances high-order, non-splitting simulation of multidimensional detonation waves by combining the space-time adaptive $ADER$-$DG$ framework with a localized $LST$-$DG$ predictor and an a posteriori sub-cell $ADER$-$WENO$ limiter. It introduces a per-cell, stiffness-aware time-step partition that obviates global adaptive stepping, enabling accurate resolution of ZND detonation structures on relatively coarse meshes. The approach is validated across a spectrum of canonical gas-dynamics tests and then applied to plane, cylindrical, and spherical detonations, as well as detonation interactions with inert inhomogeneities, illustrating subcell resolution, minimal nonphysical artifacts, and robust performance in highly nonlinear, multiscale reactive flows. The results demonstrate that the method can reproduce the essential features of detonation waves, including the Zel'dovich peak and complex cellular structures, with high fidelity in 2D and 3D settings, offering a practical tool for engineering and physical analysis of reacting flows. Overall, the work shows that space-time adaptive $ADER$-$DG$ with a posteriori sub-cell limiting is well-suited for challenging detonation problems in heterogeneous media, with potential impact on simulations of real multi-component fuels and explosives.

Abstract

The space-time adaptive ADER-DG finite element method with LST-DG predictor and a posteriori sub-cell ADER-WENO finite-volume limiting was used for simulation of multidimensional reacting flows with detonation waves. The presented numerical method does not use any ideas of splitting or fractional time steps methods. The modification of the LST-DG predictor has been developed, based on a local partition of the time step in cells in which strong reactivity of the medium is observed. This approach made it possible to obtain solutions to classical problems of flows with detonation waves and strong stiffness, without significantly decreasing the time step. The results obtained show the very high applicability and efficiency of using the ADER-DG-PN method with a posteriori sub-cell limiting for simulating reactive flows with detonation waves. The numerical solution shows the correct formation and propagation of ZND detonation waves. The structure of detonation waves is resolved by this numerical method with subcell resolution even on coarse spatial meshes. The smooth components of the numerical solution are correctly and very accurately reproduced by the numerical method. Non-physical artifacts of the numerical solution, typical for problems with detonation waves, such as the propagation of non-physical shock waves and weak detonation fronts ahead of the main detonation front, did not arise in the results obtained. The results of simulating rather complex problems associated with the propagation of detonation waves in significantly inhomogeneous domains are presented, which show that all the main features of detonation flows are correctly reproduced by this numerical method. It can be concluded that the space-time adaptive ADER-DG-PN method with a posteriori sub-cell ADER-WENO finite-volume limiting is perfectly applicable to simulating complex reacting flows with detonation waves.

Figures (36)

• Figure 1: Numerical solution of the three-dimensional problem of an isentropic vortex advection (a detailed statement of the problem is presented in the text) obtained using the ADER-DG-$\mathbb{P}_{12}$ method on single cell mesh $1 \times 1 \times 1$ at the final time $t_{\rm final} = 10.0$: subcells finite-volume representation of exact (left) and numerical (right) solutions for density $\rho$ (top) and velocity $\mathbf{v}$ (bottom).
• Figure 2: Numerical solution of the classical Sod, Lax, two rarefaction waves and two shock waves problems (from top to bottom) obtained using the ADER-DG-$\mathbb{P}_{5}$ method with a posteriori limitation of the solution by a ADER-WENO2 finite volume limiter on mesh with $200$ cells (a detailed statement of the problem is presented in the text). The graphs show the coordinate dependencies of density $\rho$, pressure $p$, flow velocity $u$ and troubled cells indicator $\beta$ (from left to right) 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 two-dimensional Kelvin-Helmholtz instability problem (a detailed statement of the problem is presented in the text) obtained using the ADER-DG-$\mathbb{P}_{2}$ (left) and ADER-DG-$\mathbb{P}_{9}$ (right) methods on mesh with $100 \times 100$ cells at the time $t = 0.8$. The graphs show the coordinate dependencies of the subcells finite-volume representation of density $\rho$ (top) and troubled cells indicator $\beta$ (bottom).
• Figure 4: Numerical solution of the two-dimensional Kelvin-Helmholtz instability problem (a detailed statement of the problem is presented in the text) obtained using the ADER-DG-$\mathbb{P}_{2}$ (left) and ADER-DG-$\mathbb{P}_{9}$ (right) methods on mesh with $100 \times 100$ cells at the time $t = 2.0$. The graphs show the coordinate dependencies of the subcells finite-volume representation of density $\rho$ (top) and troubled cells indicator $\beta$ (bottom).
• Figure 5: Numerical solution of the classical two-dimensional Riemann problems RP1, RP2, RP3, RP4 and RP5 (from top to bottom) obtained using the ADER-DG-$\mathbb{P}_{2}$ method (left two columns) and the ADER-DG-$\mathbb{P}_{5}$ method (right two columns) methods with a posteriori limitation of the solution by a ADER-WENO2 finite volume limiter on mesh with $100 \times 100$ cells (a detailed statement of the problem is presented in the text). The graphs show the coordinate dependencies of density $\rho$ (first and third columns) and troubled cells indicator $\beta$ (second and fourth columns) at the final time $t_{\rm final} = 0.25$ (RP1, RP2, RP4, RP5) and $0.30$ (RP3).