A new, transformed flux-form semi-Lagrangian (TFSL) scheme provides stable and accurate algorithm to solve the advection-diffusion equation. Different from the existing flux-form semi-Lagrangian schemes, the flux at boundary of grid cell is treated as the temporal mean between present and next time steps. After the temporal-spatial transformation using the characteristic-line concept, the temporal integration of the flux from present to next time step becomes the spatial integration of the flux at the present time step. This scheme is always stable even for large Courant numbers (> 20) with the second order accuracy in both time and space. Capability of the TFSL scheme is demonstrated by comparison to the existing schemes such as the upwind, central, and Lax-Wendroff explicit schemes in simulating propagation of the equatorial Rossby soliton (a standard ocean modeling test case).