We discuss deriving numerical models by directly discretizing the action principle. This preserves, insofar as possible, the connection between symmetries and conservation laws, leading, empirically, to superior performance. We will present examples related to fluid and kinetic models of laser-plasma interactions.
Routinely numerical models (such as those used to model laser-plasma interactions) are obtained by discretizing, either through means of a grid or a basis expansion, the relevant equations of motion. The primarily difficulty with this approach has to do with the effects of discretization on the system's conservation laws. In effect, the numerical discretization has the character of an uncontrolled approximation. When the underlying dynamics satisfies an action principle, there appears to be a significantly better alternative: performing the discretization directly in the action. This preserves the link between symmetries and conservation laws and, empirically, leads to better behaviour compared with models obtained from an ad-hoc discretization. Since this approach yield a finite degree-of-freedom Lagrangian systems, in general, it is possible to perform a Legendre transform to obtain an (canonical) Hamiltonian system.
Our recent variational formulation of macro-particle models [1-3] is an example of this general approach. Nothing in the approximations introduces explicit time dependence to the Lagrangian and thus the continuous-time equations of motion exactly conserve energy. In addition, the variational formulation allows for constructing models of arbitrary spatial and temporal order. In contrast, the overall accuracy of the usual PIC algorithm is at most second due to the nature of the force interpolation between the gridded field quantities and the particle position. Here the macro-particle shape is arbitrary; the spatial extent is completely decoupled from both the grid-size and the "smoothness" of the shape. For the electromagnetic case , gauge invariance and momentum conservation are considered in detail and it is shown that using a truncated Fourier representation allows for the simultaneous conservation of energy and momentum. The requirements for exact invariance are explored and it is shown that one viable option is to represent the potentials with a truncated Fourier basis.
 E. G. Evstatiev and B. A. Shadwick, JCP 245, 376 (2013).
 A. B. Stamm, B. A. Shadwick, and E. G. Evstatiev, IEEE TPS 42, 1747 (2014)
 B. A. Shadwick, A. B. Stamm, and E. G. Evstatiev, Phys. Plasmas 21, 055708 (2014).
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