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The accurate modelling of trajectories in accelerators, in particular for long simulation times or large particle amplitudes necessitate the employment of so called "symplectic" integration schemes. These schemes preserve the conservative nature of motion, i.e. the energy, as defined through the Hamiltonian function. For the accelerator Hamiltonian, it is possible to employ a new set of high order symplectic integrators (CSABAm). One benefit of these integrators is their strong numerical stability, which results from the inclusion of only forward integration steps, independent of the order of accuracy. The power of these integrators, with respect to accuracy and computational time, is demonstrated in simulations of simple lattices and benchmarked against other well established integration schemes in the accelerator community. A concrete application is then presented, namely the effectiveness of simple DC wires in order to compensate the detrimental effect of beam-beam long-range interactions in the High Luminosity Large Hadron Collider. Based on the estimation of beam dynamics indicators correlated with beam lifetime, such as dynamic aperture and frequency map analysis, it is shown how these long range beam-beam interactions can be very well mitigated, for different scenarios of the HL-LHC, with a proper choice of their current and distance from the beam, while respecting machine protection restrictions.