  Spacetime discontinuous Galerkin (SDG) finite element methods are used to
solve such PDEs involving space and time variables arising from wave
propagation phenomena in important applications in science and engineering.
  To support an accurate and efficient solution procedure using SDG methods and
to exploit the flexibility of these methods, we give a meshing algorithm to
construct an unstructured simplicial spacetime mesh over an arbitrary
simplicial space domain. Our algorithm is the first spacetime meshing algorithm
suitable for efficient solution of nonlinear phenomena in anisotropic media
using novel discontinuous Galerkin finite element methods for implicit
solutions directly in spacetime. Given a triangulated d-dimensional Euclidean
space domain M (a simplicial complex) and initial conditions of the underlying
hyperbolic spacetime PDE, we construct an unstructured simplicial mesh of the
(d+1)-dimensional spacetime domain M x [0,infinity). Our algorithm uses a
near-optimal number of spacetime elements, each with bounded temporal aspect
ratio for any finite prefix M x [0,T] of spacetime. Our algorithm is an
advancing front procedure that constructs the spacetime mesh incrementally, an
extension of the Tent Pitcher algorithm of Ungor and Sheffer (2000).
  In 2DxTime, our algorithm simultaneously adapts the size and shape of
spacetime tetrahedra to a spacetime error indicator. We are able to incorporate
more general front modification operations, such as edge flips and limited mesh
smoothing. Our algorithm represents recent progress towards a meshing algorithm
in 2DxTime to track moving domain boundaries and other singular surfaces such
as shock fronts.
