A conservative nodal variational multiscale method for Lagrangian shock hydrodynamics

A new method based on a continuous, piece-wise linear approximation of the equations for Lagrangian shock hydrodynamics is presented. Numerical instabilities are controlled by a stabilizing operator derived using the paradigm of the variational multiscale analysis. Encouraging numerical comparisons with existing methods in the case of quadrilateral and hexahedral elements indicate that the proposed method is capable of preventing hourglass patterns in the solution, while maintaining accuracy in regions of smooth flow. The proposed approach satisfies Galilean invariance properties and hinges upon the interpretation of the Lagrangian shock hydrodynamics equations as a system of nonlinear wave equations. A specific implementation in terms of a predictor/multi-corrector version of the mid-point time integrator guarantees global conservation of mass, momentum, and total energy for each iterate. Stability and formal order of accuracy are investigated applying the von Neumann analysis to the linearized shock hydrodynamics equations in one dimension. This approach yields tight bounds for stable time-step estimation, formal second-order accuracy of the method in time and space, and valuable indications on the choice of the most appropriate values for the stabilization parameters present in the formulation.