TNSA proton maximum energy laws for 2D and 3D PIC simulations

Highlights: • Two empirical laws for maximum energy in 2D and 3D sims, depending on $E_{\infty }$. • Fits from 2D and 3D results are quite good and the extrapolated values are comparable. • The fitting appears to be satisfactory also for small incidence angles. • The model can avoid the arbitrariness in the choice of the stop time for PIC runs. • 2D simulations may have a quantitative value, rather than being purely qualitative. Numerical simulations are a prominent tool in laser-plasma experiments. Their role, as a guide in new regime explorations and as a support for understanding laboratory results, is undisputed. But as the experiments themselves are growing in costs, setup time and complexity, so are the numerical counterparts. Nowadays it is often necessary to investigate, with great accuracy, a huge set of parameters, in order to explore interesting features. In literature, it is well known that two-dimensional particle in cell (2D PIC) simulations can only give a qualitative estimation of experimental results, often through a great layer of arbitrariness. On the other hand, three-dimensional (3D) PIC simulations, for the same setup, can typically require two orders of magnitude more of computational resources, to deliver results that, while being similar to laboratory results, are still far from being a real match, due to the many uncertainties, in the parameters and in the model, included in the physical engine. Following our recent published work, we discuss here a couple of empirical laws that we proposed, that can help giving quantitative insight into 2D PIC simulations and determining when 3D simulations should be stopped, if it is not really necessary to do a detailed exploration of the numerical results at long times.