  We describe the remapped particle-mesh method, a new mass-conserving method
for solving the density equation which is suitable for combining with
semi-Lagrangian methods for compressible flow applied to numerical weather
prediction. In addition to the conservation property, the remapped
particle-mesh method is computationally efficient and at least as accurate as
current semi-Lagrangian methods based on cubic interpolation. We provide
results of tests of the method in the plane, results from incorporating the
advection method into a semi-Lagrangian method for the rotating shallow-water
equations in planar geometry, and results from extending the method to the
surface of a sphere.
