The Particle Finite Element Method-Second Generation (PFEM-2) is an alternative PFEM technique to solve the incompressible Navier-Stokes problem with large time steps. The method was presented very recently in . The key idea of the method is based on coupling the standard PFEM with the X-IVAS (eXplicit Integration following the Velocity and Acceleration Streamlines) method, which consists of integrating the convective terms following the streamlines rather than the particle trajectories.
Left: PFEM-2 with moving mesh. Right: PFEM-2 with fixed mesh.
PFEM-2 Simulation of the Rayleigh-Taylor instability at Atwood number 3. Comparison among simulations with different time-steps 
Experimental validation of PFEM-2. Case: collapse of a water column 
A series of combined-explicit implicit methods based on employing PFEM2 strategy in a fully Lagrangian framework (actually moving grid + virtually moving grid) was proposed in [11,12]. The methodology allowed obtaining effecient schemes not bound by severe time step limitations and, at the same time, characterized by second-order accuracy in time [11,12].
An interesting feature of PFEM-2 is the possibility to use an explicit time integration independently on the Courant number. The method remains explicit and stable independently on the mesh size. The time step is established following only accuracy considerations, besides the limits given by the Fourier number. In  a second-order accurate in time and space formulation showed drastic computing times savings regarding standard Eulerian alternatives.
Injection of a liquid jet in a chamber with PFEM-2. Tracking of the main core and primary atomization 
The PFEM-2 has been applied successfully to different engineering problems. In , the ideas presented in [1, 2] were generalized for multifluid flows with large time steps.
PFEM-2 simulation of the Rayleigh-Taylor instability at Atwood number 3 
In , an extended validation of the method for academic problems is presented. Gimenez et al.  show the potential of PFEM-2 to simulate industrial problems of large time duration. An application of the method to jet atomization simulation can be found in . Becker and Idelsohn  show the potential of PFEM-2 to simulate large scale landslides events. FSI problems are tackled with a monolithic PFEM-2 approach in . Finally,  shows an application of the method to the simulation of sediment transport phenomena in rivers.
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