  Partial differential equations (PDEs) on surfaces arise in a wide range of
applications. The closest point method (Ruuth and Merriman, J. Comput. Phys.
227(3):1943-1961, [2008]) is a recent embedding method that has been used to
solve a variety of PDEs on smooth surfaces using a closest point representation
of the surface and standard Cartesian grid methods in the embedding space. The
original closest point method (CPM) was designed for problems posed on static
surfaces, however the solution of PDEs on moving surfaces is of considerable
interest as well. Here we propose solving PDEs on moving surfaces using a
combination of the CPM and a modification of the grid based particle method
(Leung and Zhao, J. Comput. Phys. 228(8):2993{3024, [2009]). The grid based
particle method (GBPM) represents and tracks surfaces using meshless particles
and an Eulerian reference grid. Our modification of the GBPM introduces a
reconstruction step into the original method to ensure that all the grid points
within a computational tube surrounding the surface are active. We present a
number of examples to illustrate the numerical convergence properties of our
combined method. Experiments for advection-diffusion equations that are
strongly coupled to the velocity of the surface are also presented.
