Numerical simulations for tires are challenging. They involve large deformations models in highly heterogeneous structures, made of non-linear materials. And impact simulation adds fast transient dynamics. An explicit time-integrator is then well suited. Non-linear stresses are easily estimated with the explicit displacement, making computation cost low. At contact zone, stresses are particularly high. They can even lead to rupture. Therefore their computation must be very accurate. Contact formulation with Lagrange multipliers are then preferred to penalty methods. The Non-Smooth Contact Dynamic (NSCD) framework provides robust algorithms. They deal with velocity discontinuities at impact without adjusting time-step. CD-Lagrange scheme presents these properties. For time-integration, it uses the explicit central difference method. It keeps the symplectic properties of this method: energy and angular momentum conservation. This is required for accuracy in non-linear large rotations \cite{DiStasio}. For contact condition, it uses Lagrange multipliers on velocity. This formulation is more stable than classic formulations on displacement, and present a better energy balance. But an issue remains for the contact problem. It is generally a Linear Complementary Problem (LCP) which links all degrees of freedom at contact zone. This work investigates how to simplify this LCP, or solve it fast. A simpler contact problem is obtained using mass redistribution methods on CD-Lagrange scheme. This new scheme keeps the conservation properties of classic CD-Lagrange. And for classic CD-Lagrange, new LCP solving methods are proposed more suited to explicit time-integration.