An $rp$-adaptive method for accurate resolution of shock-dominated viscous flow based on implicit shock tracking

Abstract

This work introduces an optimization-based $rp$-adaptive numerical method to approximate solutions of viscous, shock-dominated flows using implicit shock tracking and a high-order discontinuous Galerkin discretization on traditionally coarse grids without nonlinear stabilization (e.g., artificial viscosity or limiting). The proposed method adapts implicit shock tracking methods, originally developed to align mesh faces with solution discontinuities, to compress elements into viscous shocks and boundary layers, functioning as a novel approach to aggressive $r$-adaptation. This form of $r$-adaptation is achieved naturally as the minimizer of the enriched residual with respect to the discrete flow variables and coordinates of the nodes of the grid. Several innovations to the shock tracking optimization solver are proposed to ensure sufficient mesh compression at viscous features to render stabilization unnecessary, including residual weighting, step constraints and modifications, and viscosity-based continuation. Finally, $p$-adaptivity is used to locally increase the polynomial degree with three clear benefits: (1) lessens the mesh compression requirements near shock waves and boundary layers, (2) reduces the error in regions where $r$-adaptivity is not sufficient with the given grid topology, and (3) reduces computational cost by performing a majority of the $r$-adaptivity iterations on the coarsest discretization. A series of numerical experiments show the proposed method effectively resolves viscous, shock-dominated flows, including accurate prediction of heat flux profiles produced by hypersonic flow over a cylinder, and compares favorably in terms of accuracy per degree of freedom to $h$-adaptation with a high-order discretization.

Figures (35)

• Figure 1: Elementwise magnitude of the enriched DG residual ($\Delta = 2$) for $M_\infty=5$, $\mathrm{Re}=10^3$ hypersonic flow over a cylinder (isothermal wall) without (left) and with (right) boundary layer scaling ($\lambda$ = 10).
• Figure 2: Distribution of the Young's modulus field $E$ based on the inverse volume of reference mesh zahr_implicit_2020 (left) and the enriched residual (right) for $M_\infty=5$, $\mathrm{Re}=10^3$ hypersonic flow over a cylinder (prior to $p$-adaptation). The Young's modulus are provided on the physical domain (top) and reference domain (bottom) for clarity.
• Figure 3: Mesh and solution (first-order finite volume with $\nu=10^{-3}$) used to initialize HOIST method (pure $r$-adaptation) for viscous Burgers' problem with straight shock.
• Figure 4: HOIST solution (pure $r$-adaptation) in the physical domain $\mathcal{G}(\Omega_0)$ to the viscous Burgers' problem with straight shock at $\nu = 10^{-3}, 5\times 10^{-4}, 10^{-4}$ (left-to-right) with and without mesh edges shown. Colorbar in Figure \ref{['fig:vburg0_sptm_ic']}.
• Figure 5: HOIST solution (pure $r$-adaptation) in the reference domain $\Omega_0$ to the viscous Burgers' problem with straight shock at $\nu = 10^{-3}, 5\times 10^{-4}, 10^{-4}$ (left-to-right) with and without mesh edges shown. Colorbar in Figure \ref{['fig:vburg0_sptm_ic']}.

Theorems & Definitions (5)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5