Numerical techniques for solving the quantum constraint equation of generic lattice-refined models in loop quantum cosmology

To avoid instabilities in the continuum semiclassical limit of loop quantum cosmology models, refinement of the underlying lattice is necessary. This lattice refinement leads to new dynamical difference equations which, in general, do not have a uniform step size, implying complications in their analysis and solutions. We propose a numerical method based on Taylor expansions, which can give us the necessary information to calculate the wave function at any given lattice point. The method we proposed can be applied in any lattice-refined model, while in addition the accuracy of the method can be estimated. Moreover, we confirm numerically the stability criterion which was earlier found following a von Neumann analysis. Finally, the ''motion'' of the wave function due to the underlying discreteness of the space-time is investigated, for both a constant lattice, as well as lattice refinement models.