We describe an adaptive *hp*-refinement local finite element procedure
for the parallel solution of hyperbolic conservation laws on rectangular
domains. The local finite element procedure utilizes spaces of
piecewise-continuous polynomials of arbitrary degree and coordinated explicit
Runge-Kutta temporal integration. A solution limiting procedure produces
monotonic solutions near discontinuities while maintaining high-order
accuracy near smooth extrema. A modified tiling procedure maintains processor
load balance on parallel, distributed-memory MIMD computers by migrating
finite elements between processors in overlapping neighborhoods to produce
locally balanced computations. Grids are stored in tree data structures, with
finer grids being offspring of coarser ones. Within each grid, AVL trees
simplify the transfer of information between neighboring processors and the
insertion and deletion of elements as they migrate between processors.
Computations involving Burgers' and Euler's equations of inviscid flow
demonstrate the effectiveness of the *hp*-refinement and balancing
procedures relative to non-balanced adaptive and balanced non-adaptive
procedures.