  We present a general concept to accelerate non-relativistic charged
particles. Our concept employs an adiabatically-tapered dielectric-lined
waveguide which supports accelerating phase velocities for synchronous
acceleration. We propose an ansatz for the transient field equations, show it
satisfies Maxwell's equations under an adiabatic approximation and find
excellent agreement with a finite-difference time-domain computer simulation.
The fields were implemented into the particle-tracking program {\sc astra} and
we present beam dynamics results for an accelerating field with a
1-mm-wavelength and peak electric field of 100~MV/m. The numerical simulations
indicate that a $\sim 200$-keV electron beam can be accelerated to an energy of
$\sim10$~MeV over $\sim 10$~cm. The novel scheme is also found to form electron
beams with parameters of interest to a wide range of applications including,
e.g., future advanced accelerators, and ultra-fast electron diffraction.
