  This paper is concerned with moving mesh finite difference solution of
partial differential equations. It is known that mesh movement introduces an
extra convection term and its numerical treatment has a significant impact on
the stability of numerical schemes. Moreover, many implicit second and higher
order schemes, such as the Crank-Nicolson scheme, will loss their unconditional
stability. A strategy is presented for developing temporally high order,
unconditionally stable finite difference schemes for solving linear
convection-diffusion equations using moving meshes. Numerical results are given
to demonstrate the theoretical findings.
