Abstract: A large time step wave adding scheme originated from LeVeque's large time step Godunov scheme was presented, i.e. constructing numerical schemes via adding strong waves of discontinuity decomposition. Compared with the original scheme, a different strategy for wave adding was presented and extended to multi-dimensional cases. For rarefaction waves, a grid cell decomposition method which can automatically satisfy the entropy condition and avoid nonphysical solutions was proposed. The detailed formulae of the scheme were given, and numerical experiments using scalar equations and Euler equations in one and multi-dimensional cases were carried out. Numerical results show that, besides the advantage of large time step, the new scheme has a lower numerical dissipation and a higher resolution of shocks and contact discontinuities with the increase of CFL number in a certain range.