  A spatially discrete version of the general kink-bearing nonlinear
Klein-Gordon model in (1+1) dimensions is constructed which preserves the
topological lower bound on kink energy. It is proved that, provided the lattice
spacing h is sufficiently small, there exist static kink solutions attaining
this lower bound centred anywhere relative to the spatial lattice. Hence there
is no Peierls-Nabarro barrier impeding the propagation of kinks in this
discrete system. An upper bound on h is derived and given a physical
interpretation in terms of the radiation of the system. The construction, which
works most naturally when the nonlinear Klein-Gordon model has a squared
polynomial interaction potential, is applied to a recently proposed continuum
model of polymer twistons. Numerical simulations are presented which
demonstrate that kink pinning is eliminated, and radiative kink deceleration
greatly reduced in comparison with the conventional discrete system. So even on
a very coarse lattice, kinks behave much as they do in the continuum. It is
argued, therefore, that the construction provides a natural means of
numerically simulating kink dynamics in nonlinear Klein-Gordon models of this
type. The construction is compared with the inverse method of Flach, Zolotaryuk
and Kladko. Using the latter method, alternative spatial discretizations of the
twiston and sine-Gordon models are obtained which are also free of the
Peierls-Nabarro barrier.
