The equations that govern the behavior of physical systems can often be numerically solved using finite-difference (or finite-element) discretizations and differential-algebraic equation (DAE) solvers. For example, such an approach can be used to solve the Einstein field equations of general relativity, and thereby simulate significant astrophysical events. Fluid flow, heat transfer and other problems can also be formulated and numerically solved as DAEs. In each case, the differential equations describe the flow or motion of the physical system, and the algebraic equations describe various conservation laws, boundary conditions or constraints.
Fortunately, high-quality mathematical software for solving DAEs exists (e.g., DASSL) and it can be used to solve a wide array of difficult problems. This talk describes how DAE codes such as DASSL can be enhanced in a way that enables scientists to efficiently solve a larger range of important, multidimensional initial-value problems. In particular, we describe some preliminary work in which the gravitational field of a black hole is computed by solving some simplified versions of the Einstein field equations.