Multi-Dimensional Astrophysical Structural and Dynamical Analysis I. Development of a Nonlinear Finite Element Approach

A new field of numerical astrophysics is introduced which addresses the solution of large, multidimensional structural or slowly-evolving problems (rotating stars, interacting binaries, thick advective accretion disks, four dimensional spacetimes, etc.), as well as rapidly-evolving systems. The technique employed is the Finite Element Method (FEM), which has been used to solve engineering structural problems for more than three decades. The approach developed herein has the following key features: 1. The computational mesh can extend into the time dimension, as well as space — generally only a few cells deep for most (flat-space) astrophysical problems, but throughout spacetime for solving Einstein’s field equations. 2. When time is treated as a mesh dimension, virtually all equations describing the astrophysics of continuous media, including the field equations, can be written in a compact form similar to that routinely solved by most engineering finite element codes (albeit for nonlinear equations in a four-dimensional spacetime instead of linear ones in two or three space dimensions): the divergence of a generalized stress tensor equals a generalized body force vector, both of which are functions only of position, the state variables and their gradients. 3. The transformations that occur naturally in the four-dimensional FEM possess both coordinate and boost features, such that (a) although the computational mesh may have a complex, non-analytic, curvilinear structure, and may be adapted to the geometry of the problem, the physical equations still can be written in a simple coordinate system that is independent of the mesh structure. (b) if the mesh has a complex flow velocity with respect to coordinate space, the transformations will form the proper advective derivatives, automatically converting the equations to arbitrary Lagrangian-Eulerian. 4. Only relatively simple differential equations need to be encoded. The complex difference equations on the arbitrary curvilinear grid are generated automatically by the FEM integrals. A different integration method must be used for equations of odd and even order.