  Solution interpolation between deforming meshes is an important component for
several applications in scientific computing, including indirect
arbitrary-Lagrangian-Eulerian and rezoning moving mesh methods in numerical
solution of partial differential equations. In this paper, a high-order,
conservative, and positivity-preserving interpolation scheme is developed based
on the discontinuous Galerkin solution of a linear time-dependent equation on
deforming meshes. The scheme works for bounded but otherwise arbitrary mesh
deformation from the old mesh to the new one. The cost and positivity
preservation (with a linear scaling limiter) of the DG-interpolation are
investigated. Numerical examples are presented to demonstrate the properties of
the interpolation scheme. The DG-interpolation is applied to the rezoning
moving mesh DG solution of the radiative transfer equation, an
integro-differential equation modeling the conservation of photons and
involving time, space, and angular variables. Numerical results obtained for
examples in one and two spatial dimensions with various settings show that the
resulting rezoning moving mesh DG method maintains the same convergence order
as the standard DG method, is more efficient than the method with a fixed
uniform mesh, and is able to preserve the positivity of the radiative
intensity.
