  The Landau-Lifshitz equation provides an efficient way to account for the
effects of radiation reaction without acquiring the non-physical solutions
typical for the Lorentz-Abraham-Dirac equation. We solve the Landau-Lifshitz
equation in its covariant four-vector form in order to control both the energy
and momentum of radiating particle. Our study reveals that implicit
time-symmetric collocation methods of the Runge-Kutta-Nystr\"om type are
superior in both accuracy and better maintaining the mass-shell condition than
their explicit counterparts. We carry out an extensive study of numerical
accuracy by comparing the analytical and numerical solutions of the
Landau-Lifshitz equation. Finally, we present the results of simulation of
particles scattering by a focused laser pulse. Due to radiation reaction,
particles are less capable for penetration into the focal region, as compared
to the case of radiation reaction neglected. Our results are important for
designing the forthcoming experiments with high intensity laser fields.
