An Improved Integration Scheme for Mode-Coupling-Theory Equations

Within the mode-coupling theory (MCT) of the glass transition, we reconsider the numerical schemes to evaluate the MCT functional. Here we propose nonuniform discretizations of the wave number, in contrast to the standard equidistant grid, in order to decrease the number of grid points without losing accuracy. We discuss in detail how the integration scheme on the new grids has to be modified from standard Riemann integration. We benchmark our approach by solving the MCT equations numerically for mono-disperse hard disks and hard spheres and by computing the critical packing fraction and the nonergodicity parameters. Our results show that significant improvements in performance can be obtained employing a nonuniform grid.

Automated summary

In this study, the numerical schemes for evaluating the MCT functional within the glass transition theory were reconsidered. Nonuniform discretizations of wave number were proposed to decrease the number of grid points without losing accuracy. The modified integration scheme on the new grids showed significant performance improvements when solving the MCT equations for mono-disperse hard disks and hard spheres.