A structure-preserving Lagrangian discontinuous Galerkin method using flux and slope limiting

TL;DR

The paper develops a structure-preserving, Lagrangian nodal DG scheme for multi-dimensional gas dynamics by integrating a flux-corrected transport framework with a slope limiter. The method preserves positivity of specific volume, remains GCL-consistent, and attains second-order accuracy for cell-averaged variables, aided by a multidirectional Riemann solver and SSP time integration. It demonstrates robust shock-capturing and stability on classic 2D and 3D tests, including Taylor-Green, Sedov, Noh, and triple-point problems. The work offers a practical, conservation-friendly approach for high-fidelity Lagrangian hydrodynamics with potential extensions to higher-order and curvilinear elements.

Abstract

We introduce a Lagrangian nodal discontinuous Galerkin (DG) cell-centered hydrodynamics method for solving multi-dimensional hyperbolic systems. By incorporating an adaptation of Zalesak's flux-corrected transport algorithm, we combine a first-order positivity-preserving scheme with a higher-order target discretization. This results in a flux-corrected Lagrangian DG scheme that ensures both global positivity preservation and second-order accuracy for the cell averages of specific volume. The correction factors for flux limiting are derived from specific volume and applied to all components of the solution vector. We algebraically evolve the volumes of mesh cells using a discrete version of the geometric conservation law (GCL). The application of a limiter to the GCL fluxes is equivalent to moving the mesh using limited nodal velocities. Additionally, we equip our method with a locally bound-preserving slope limiter to effectively suppress spurious oscillations. Nodal velocity and external forces are computed using a multidirectional approximate Riemann solver to maintain conservation of momentum and total energy in vertex neighborhoods. Employing linear finite elements and a second-order accurate time integrator guarantees GCL consistency. The results for standard test problems demonstrate the stability and superb shock-capturing capabilities of our scheme.

Figures (5)

• Figure 1: Geometrical entities attached to element $\omega_c$.
• Figure 2: Taylor-Green vortex problem, velocity magnitude fields at different times $t$ for various mesh resolutions. Note that the solver outputs the subcells associated with each element. See Fig. \ref{['fig:geom']} for details.
• Figure 3: Three-dimensional Sedov blast wave problem, density profiles $\varrho$ at $t=1.0$ for various mesh resolutions. Note that the solver outputs the subcells associated with each element. See Fig. \ref{['fig:geom']} for details.
• Figure 4: Noh problem, density profiles $\varrho$ obtained at $t=0.6$ on a $30\times 30\times 30$ mesh. Note that the solver outputs the subcells associated with each element. See Fig. \ref{['fig:geom']} for details.
• Figure 5: Triple point problem, density profile $\varrho$ and mesh at time $t=4.0$ using $E_h=140\times 80$ elements. Note that the solver outputs the subcells associated with each element. See fig.\ref{['fig:geom']} for details.

Theorems & Definitions (8)

• Remark 3.1
• Remark 5.1
• Remark 5.2
• Remark 5.3
• Remark 5.4
• Remark 6.1
• Remark 6.2
• Remark