  We present a new high-order finite volume reconstruction method for
hyperbolic conservation laws. The method is based on a piecewise cubic
polynomial which provides its solutions a fifth-order accuracy in space. The
spatially reconstructed solutions are evolved in time with a fourth-order
accuracy by tracing the characteristics of the cubic polynomials. As a result,
our temporal update scheme provides a significantly simpler and computationally
more efficient approach in achieving fourth order accuracy in time, relative to
the comparable fourth-order Runge-Kutta method. We demonstrate that the
solutions of PCM converges in fifth-order in solving 1D smooth flows described
by hyperbolic conservation laws. We test the new scheme in a range of numerical
experiments, including both gas dynamics and magnetohydrodynamics applications
in multiple spatial dimensions.
