Generalized, energy-conserving numerical simulations of particles in general relativity. II. Test particles in electromagnetic fields and GRMHD

Direct observations of compact objects, in the form of radiation spectra, gravitational waves from VIRGO/LIGO, and forthcoming direct imaging, are currently one of the primary source of information on the physics of plasmas in extreme astrophysical environments. The modeling of such physical phenomena requires numerical methods that allow for the simulation of microscopic plasma dynamics in presence of both strong gravity and electromagnetic fields. In Bacchini et al. (2018) we presented a detailed study on numerical techniques for the integration of free geodesic motion. Here we extend the study by introducing electromagnetic forces in the simulation of charged particles in curved spacetimes. We extend the Hamiltonian energy-conserving method presented in Bacchini et al. (2018) to include the Lorentz force and we test its performance compared to that of standard explicit Runge-Kutta and implicit midpoint rule schemes against analytic solutions. Then, we show the application of the numerical schemes to the integration of test particle trajectories in general relativistic magnetohydrodynamic (GRMHD) simulations, by modifying the algorithms to handle grid-based electromagnetic fields. We test this approach by simulating ensembles of charged particles in a static GRMHD configuration obtained with the Black Hole Accretion Code (BHAC)

Symplectic Discretization Approach for Developing New Proximal Point Algorithms

Proximal point algorithms have found numerous applications in the field of convex optimization, and their accelerated forms have also been proposed. However, the most commonly used accelerated proximal point algorithm was first introduced in 1967, and recent studies on accelerating proximal point algorithms are relatively scarce. In this paper, we propose high-resolution ODEs for the proximal point operators for both closed proper convex functions and maximally monotone operators, and present a Lyapunov function framework to demonstrate that the trajectories of our high-resolution ODEs exhibit accelerated behavior. Subsequently, by symplectically discretizing our high-resolution ODEs, we obtain new proximal point algorithms known as symplectic proximal point algorithms. By decomposing the continuous-time Lyapunov function into its elementary components, we demonstrate that symplectic proximal point algorithms possess $O(1/k^2)$ convergence rates

Variational Formulation of Macro-Particle Models for Electromagnetic Plasma Simulations

A variational method is used to derive a self-consistent macro-particle model for relativistic electromagnetic kinetic plasma simulations. Extending earlier work [E. G. Evstatiev and B. A. Shadwick, J. Comput. Phys., vol. 245, pp. 376-398, 2013], the discretization of the electromagnetic Low Lagrangian is performed via a reduction of the phase-space distribution function onto a collection of finite-sized macro-particles of arbitrary shape and discretization of field quantities onto a spatial grid. This approach may be used with both lab frame coordinates or moving window coordinates; the latter can greatly improve computational efficiency for studying some types of laser-plasma interactions. The primary advantage of the variational approach is the preservation of Lagrangian symmetries, which in our case leads to energy conservation and thus avoids difficulties with grid heating. Additionally, this approach decouples particle size from grid spacing and relaxes restrictions on particle shape, leading to low numerical noise. The variational approach also guarantees consistent approximations in the equations of motion and is amenable to higher order methods in both space and time. We restrict our attention to the 1-1/2 dimensional case (one coordinate and two momenta). Simulations are performed with the new models and demonstrate energy conservation and low noise.Comment: IEEE Transaction on Plasma Science (TPS) Special Issue: Plenary and Invited Papers of the Pulsed Power and Plasma Science Conference (PPPS 2013

Compact phase space, cosmological constant, discrete time

We study the quantization of geometry in the presence of a cosmological constant, using a discretiza- tion with constant-curvature simplices. Phase space turns out to be compact and the Hilbert space finite dimensional for each link. Not only the intrinsic, but also the extrinsic geometry turns out to be discrete, pointing to discreetness of time, in addition to space. We work in 2+1 dimensions, but these results may be relevant also for the physical 3+1 case.Comment: 6 page

Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations

After we derive the Serre system of equations of water wave theory from a generalized variational principle, we present some of its structural properties. We also propose a robust and accurate finite volume scheme to solve these equations in one horizontal dimension. The numerical discretization is validated by comparisons with analytical, experimental data or other numerical solutions obtained by a highly accurate pseudo-spectral method.Comment: 28 pages, 16 figures, 75 references. Other author's papers can be downloaded at http://www.denys-dutykh.com