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A new hybrid scheme, called Lagrangian-Eulerian advection (LEA), that combines the advantages of the Eulerian and Lagrangian frameworks is applied to the visualization of dense representations of time-dependent vector fields. The algorithm encodes the particles into a texture that is then advected. By treating every particle equally, we can handle texture advection and dye advection within a single framework. High temporal and spatial correlation is achieved through the blending of successive frames. A combination of particle and dye advection enables the simultaneous visualization of streamlines, particle paths and streak-lines. We demonstrate various experimental techniques on several physical flow fields. The simplicity of both the resulting data structures and the implementation suggest that LEA could become a useful component of any scientific visualization toolkit concerned with the display of unsteady flows.