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

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.