High-order curvilinear finite elements for axisymmetric Lagrangian hydrodynamics

In this paper we present an extension of our general high-order curvilinear finite element approach for solving the Euler equations in a Lagrangian frame [1] to the case of axisymmetric problems. The numerical approximation of these equations is important in a number of applications of compressible shock hydrodynamics and the reduction of 3D problems with axial symmetry to 2D computations provides a significant computational advantage. Unlike traditional staggered-grid hydrodynamics (SGH) methods, which use the so-called “area-weighting” scheme, we formulate our semi-discrete axisymmetric conservation laws directly in 3D and reduce them to a 2D variational form in a meridian cut of the original domain. This approach is a natural extension of the high-order curvilinear finite element framework we have developed for 2D and 3D problems in Cartesian geometry, leading to a rescaled momentum conservation equation which includes new radial terms in the pressure gradient and artificial viscosity forces. We show that this approach exactly conserves energy and we demonstrate via computational examples that it also excels at preserving symmetry in problems with symmetric initial conditions. The results also illustrate that our computational method does not produce spurious symmetry breaking near the axis of rotation, as is the case with many area-weighted approaches.