Higher-order explicit methods for laser-plasma interactions

The evolution of a short, intense laser pulse propagating in an underdense plasma is of particular interest for laser-plasma accelerator physics and, in some circumstances, is well-modeled by the cold Maxwell-fluid equations. Solving this system using conventional second-order explicit methods in a three-dimensional simulation over experimentally-relevant configurations is prohibitively expensive. This motivated a search for more efficient numerical methods to solve the fluid equations. Explicit methods tend to suffer from stability constraints which couple the maximum allowable time step to the spatial grid size. If the dynamics of the system evolves on a time scale much larger than the constrained time step, an explicit method would require many more update cycles than is physically necessary. In these circumstances implicit methods, which tend to be unconditionally stable, may be attractive. But when physical situations (e.g., Raman processes) need to resolve the fast dynamics, implicit methods are unlikely to exhibit much improvement over explicit methods. Thus, we look for higher-order explicit methods in space that would allow coarser spatial grids and larger time steps. We restrict our discussion to the one-dimensional case and present a comprehensive survey of a wide range of numerical methods to solve the fluid equations, including methods of order two through six in space and two through eight in time. A systematic approach to determine the stability condition is presented using linear stability analysis of numerical dispersion relations. Three higher-order methods are implemented to show their behavior, in terms of numerical stability and energy conservation.