A Deferred Correction, Continuous Galerkin Method for Curvilinear Staggered-Grid Lagrangian Hydrodynamics

Abstract

We present a continuous Galerkin, Deferred Correction (cG-DeC) method for the equations of Lagrangian hydrodynamics. The proposed scheme combines a high-order continuous finite element discretization with an explicit Deferred Correction time integrator, yielding a formulation that avoids the inversion of a global sparse mass matrix at each update. To stabilize the method in the presence of strong shocks we introduce a modification of the hyperviscosity model that is compatible with the cG-DeC framework and likewise does not require a global solve to construct the finite element approximation to the polyharmonic operator. We further present an alternative reformulation of the DeC iteration which admits a simple recursive algorithmic structure and provides insight into previously observed convergence behavior of explicit DeC applied to hyperbolic partial differential equations. Conservation of momentum and total energy of the resulting fully discrete scheme is analyzed. A set of numerical experiments illustrates the accuracy and robustness of the cG-DeC scheme. Comparisons with a continuous Galerkin Runge-Kutta (cG-RK) method are provided.