Lagrangian shock hydrodynamics on tetrahedral meshes: A stable and accurate variational multiscale approach

In the past, a number of attempts have failed to robustly compute highly transient shock hydrodynamics flows on tetrahedral meshes. To a certain degree, this is not a surprise, as prior attempts emphasized enhancing the structure of shock-capturing operators rather than focusing on issues of stability with respect to small, linear perturbations. In this work, a new method is devised to stabilize computations on piecewise-linear tetrahedral finite elements. Spurious linear modes are prevented by means of the variational multiscale approach. The resulting algorithm can be proven stable in the linearized limit of acoustic wave propagation. Starting from this solid base, the approach is generalized to fully nonlinear shock computations, by augmenting the discrete formulation with discontinuity-capturing artificial viscosities. Extensive tests in the case of Lagrangian shock dynamics of ideas gases on triangular and tetrahedral grids confirm the stability and accuracy properties of the method. Incidentally, the same tests also reveal the lack of stability of current compatible/mimetic/staggered discretizations: This is due to the presence of specific unstable modes which are theoretically analyzed and verified in computations.