In this work, an efficient way to predict the dynamics of a scalar at high Schmidt numbers, advected by a turbulent flow, is presented. For high Schmidt numbers, the spatial resolution required for the scalar field has to be finer than the resolution required for the velocity field, leading to a significant computational cost due to the Courant-Friedrichs-Lewy (CFL) constraint. We propose here a remeshed particle method coupled to a spectral flow solver to overcome this computational cost limitation. This allows us to perform a systematic analysis of flows over a wide range of Reynolds and Schmidt numbers. For high enough Reynolds and Schmidt numbers, the results presented here recover the spectral behavior predicted by theory. First, the classic k−5/3 law (where k is the wave number) is found for the inertial-convective range. At intermediate scales, the viscous-convective range exhibits a k−1 law for Schmidt numbers higher than unity. Finally, the numerical results indicate that the dissipation range agree well with the Kraichnan model for high Schmidt numbers.