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In this paper, we review three methods to construct symplectic and self-consistent multiparticle algorithms to simulate space–charge effects in particle accelerators. The first method is based on a discrete multiparticle Hamiltonian with an interaction term that depends explicitly on the coordinates of the macroparticles. The second method derives from Low’s Lagrangian for a collisionless plasma. The third method is based on a corresponding collisionless Hamiltonian. The last two methods have been mostly developed by the plasma physics community, but are equally applicable to accelerator physics problems.

Computer simulations have had a profound impact on the design and understanding of past and present plasma acceleration experiments, and will be a key component for turning plasma accelerators from a promising technology into a mainstream scientific tool. In this article, we present an overview of the numerical techniques used with the most popular approaches to model plasma-based accelerators: electromagnetic particle-in-cell, quasistatic and ponderomotive guiding center. The material that is presented is intended to serve as an introduction to the basics of those approaches, and to advances (some of them very recent) that have pushed the state of the art, such as the optimal Lorentz-boosted frame, advanced laser envelope solvers and the elimination of numerical Cherenkov instability. The particle-in-cell method, which has broader interest and is more standardized, is presented in more depth. Additional topics that are cross-cutting, such as azimuthal Fourier decomposition or filtering, are also discussed, as well as potential challenges and remedies in the initialization of simulations and output of data. Examples of simulations using the techniques that are presented have been left out of this article for conciseness, and because simulation results are best understood when presented together, and contrasted with theoretical and/or experimental results, as in other articles of this volume.

Computer simulations have had a profound impact on the design and understanding of past and present plasma acceleration experiments, and will be a key component for turning plasma accelerators from a promising technology into a mainstream scientific tool. In this article, we present an overview of the numerical techniques used with the most popular approaches to model plasma-based accelerators: electromagnetic particle-in-cell, quasistatic and ponderomotive guiding center. The material that is presented is intended to serve as an introduction to the basics of those approaches, and to advances (some of them very recent) that have pushed the state of the art, such as the optimal Lorentz-boosted frame, advanced laser envelope solvers and the elimination of numerical Cherenkov instability. The particle-in-cell method, which has broader interest and is more standardized, is presented in more depth. Additional topics that are cross-cutting, such as azimuthal Fourier decomposition or filtering, are also discussed, as well as potential challenges and remedies in the initialization of simulations and output of data. Examples of simulations using the techniques that are presented have been left out of this article for conciseness, and because simulation results are best understood when presented together, and contrasted with theoretical and/or experimental results, as in other articles of this volume.