We investigate the use of Helmholtz decomposition to con- struct a fast solver for the transport equation. Our method is based on separating the dynamics into different parts with different qualitative features. For each part we adapt the numerical method to effectively exploit specific features of this part. Then the solvers are combined to a solver for the complete system. The last part can be done for exam- ple by using our solvers as preconditioners for a traditional method or in an operator splitting framework. The aim of this work is to demonstrate the use of Helmholtz decom- position in this setting.