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Numerical Methods for PDEs: In Occasion of Raytcho Lazarov's 70th Birthday

Tzanio Kolev, Lawrence Livermore National Laboratory
High-Order Curvilinear Finite Elements for Lagrangian Hydrodynamics


  • Tzanio Kolev, Lawrence Livermore National Laboratory
  • Veselin Dobrev, Lawrence Livermore National Laboratory
  • Robert Rieben, Lawrence Livermore National Laboratory


The discretization of the Euler equations of gas dynamics in a moving Lagrangian frame is at the heart of many multi-physics simulation algorithms. In this talk, we present a general framework for high-order Lagrangian discretizations of the compressible shock hydrodynamics equations using curvilinear finite elements. This method is derived through a variational formulation of the momentum and energy conservation equations using high-order continuous finite elements for the velocity and position, and a high-order discontinuous basis for the internal energy field. The use of high-order position description enables curvilinear zone geometries allowing for better approximation of the mesh curvature which develops naturally with the flow. The semi-discrete equations involve velocity and energy mass matrices which are constant in time due to the notion of strong mass conservation. We also introduce the concept of generalized corner force matrices, which together with the strong mass conservation principle, imply the exact total energy conservation on a semi-discrete level. The fully-discrete equations are obtained by the application of a Runge Kutta-like energy conserving time stepping scheme. We review the implementation of these ideas in our research codes, and present a number of two-dimensional, three-dimensional and axisymmetric computational results demonstrating the benefits of the high-order approach for Lagrangian computations.

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344.