"The success of PetaWatt (PW) laser facilities presently under construction, which aim at producing promising particle and light sources from relativistic laser-plasma interactions, will rely on the strong coupling between experiments and large-scale simulations with Particle-In-Cell (PIC) codes.Standard PIC codes currently in use fail to accurately describe these new interaction regimes because the finite difference Maxwell’s solver used to compute electromagnetic fields generates strong instabilities when particles move at relativistic velocities. At present, the mitigation of these instabilities requires the use of very high resolution, which dramatically increases the computation time, and prevents realistic 3D modeling.Our project aims at building a new generation of highly accurate PIC codes, which will enable realistic 3D simulations of these yet unexplored interaction regimes. It will use highly precise pseudo-spectral methods to solve Maxwell’s equations. Despite their accuracy, such methods have however hardly been used so far, due to their low scalability to 10,000s of cores only, which is not enough to take advantage of supercomputer architectures required for 3D modeling.To break this barrier, the main challenge to tackle is the implementation of a new and pioneering grid decomposition technique proposed by the outgoing host. This will enable for the first time a massively parallel implementation of pseudo-spectral solvers on up to a million of cores. The new PIC code will then be used to model future experiments with the PW laser of the return host and compared to standard codes.The outcome of this project will be a 3D PIC code which dramatically reduces the time-to-solution needed to solve a given problem, and hence have a huge impact on the field of ultrahigh intensity laser-plasma interaction, which is extremely active in Europe. Our general approach may also impact many other fields of computational physics where spectral solvers are used."