Particle-in-Cell (PIC) simulation is an interpolation-based method on the Newton–Maxwell (N–M) system. Its well-known drawback is its shape/interpolation functions often causing the violation of continuity equations (CEs) at mesh nodes and that of Maxwell equations (MEs) at particles' positions. Whether this drawback can be overcome by choosing/solving suitable shape/interpolation functions is of fundamental importance for the PIC simulation. Until now, these shape/interpolation functions are usually subjectively chosen and, hence, always invoke the drawback. Here, we first investigate whether these shape/interpolation functions can be self-consistently solved by considering under what condition the CEs and the MEs can be satisfied anywhere. Strict mathematical analysis reveals that strict self-consistent shape/interpolation functions are unavailable. Only few approximately self-consistent shape/interpolation functions are luckily found by some authors. This fact drives us to present another universal interpolation-free strict method on the N–M system.

The mechanism of electron acceleration by intense laser pulse interacting with an underdense plasma layer is examined by one-dimensional particle-in-cell (1D-PIC) simulations. The standard dephasing limit and the electron acceleration process are discussed briefly. A new phenomenon, of short high-quality, well-collimated return relativistic electron beam with thermal energy spread, is observed in the direction opposite to laser propagation. The process of the electron beam formation, its characteristics, and the time-history in x and px space for test electrons in the beam, are analyzed and exposed clearly. Finally, an estimate for the maximum electron energy appears in a good agreement with simulation results.