## Simulation: Accurate deterministic particle methods

Particle methods provide a unique framework for the simulation of continuous deterministic systems as described by partial differential equations (PDE). Particle methods do not require a computational mesh to numerically solve a PDE. Instead, they discretize the equation over scattered particle locations, thus providing unmatched flexibility for simulations in complex and deforming geometries and on complex-shaped surfaces.

Previously available particle methods, however, only worked on regularly distributed sets of particles, hence wasting one of the important properties of meshless methods. And even on regularly distributed particles, they suffered from a constant discretization error, preventing convergence below a certain base error level.

We have recently developed a theoretical framework for correcting the discretization errors in particle methods. This allows the systematic construction of correct particle approximations, whose error converges to zero on both regular and irregular particle distributions. Moreover, the developed theory makes explicit for the first time the connections between particle methods and grid-based finite-difference methods. We showed that finite differences can be interpreted as limit cases of certain particle methods. The new framework enables for the first time accurate simulations using irregularly distributed particles, and the demonstrated connection to finite differences could open new avenues in the theory of numerical methods.

Numerical error of a particle simulation on irregularly distributed particles or near boundaries. Dashed and dotted lines: Conventional methods are inconsistent, leading to an error that grows as the distance between particles is reduced (i.e., the resolution of the simulation is increased). Solid lines: the newly presented corrected operators show the correct convergence rate in all cases.

Numerical error of a particle simulation using particles regularly placed on a Cartesian lattice. Conventional, uncorrected methods (right plot) cannot reduce the error below a certain minimum error level, which depends on the size of the particles compared to their spacing. The newly presented corrected operators restore proper convergence of the error, allowing more accurate simulations.

B. Schrader, S. Reboux, and I. F. Sbalzarini. Discretization correction of general integral PSE operators in particle methods. *J. Comput. Phys.*, 229:4159–4182, 2010.