A new B2SOLPS5.2 transport code has been developed and implemented for the simulation of H-mode shots. A new equation system is proposed, which is equivalent to the system which was used in B2SOLPS5.0 previously. The main idea is to replace the major part of the large radial ∇B driven convective fluxes by poloidal fluxes with the same divergence both in the particle balance and in the energy balance equations. This is of special importance for the H-mode where the diffusion coefficient is strongly reduced inside the barrier and large radial convective flows are strongly undesirable from the numerical point of view. The H-mode shots of ASDEX-Upgrade and MAST have been simulated with the new version with reasonable time steps and convergence. It is demonstrated that the radial electric field inside the edge transport barrier and in the pedestal region is close to the neoclassical electric field as in previous simulations of Ohmic shots. The toroidal rotation is co-current directed as in L-mode but is significantly larger in absolute value. It is shown that the shear of the poloidal drift at the inner side of the barrier is close to the value of the shear before the transition, while inside the barrier the value of the shear is significantly bigger. This fact determines self-consistently the width of the edge transport barrier. It is demonstrated that to match the experimental density and temperature radial profiles the drop in the diffusion coefficient within the barrier needs to be significantly larger than the drop in the electron heat conductivity coefficient.For the H-mode the pedestal region usually corresponds to the collisionless regime, so several corrections were introduced into the transport coefficients to extend the applicability of the code to the plateau and banana regimes in the inner regions of the simulation domain.