Novel Particle and Hybrid Methods for Complex Systems

Particle-based methods have potentially many advantages over the traditional grid-based methods for hydrodynamic simulations of highly non-uniform systems such as high energy density applications and cosmological systems. They possess numerous attractive properties such as the exact conservation via Lagrangian formalism and ability to robustly handle material interfaces of any complexity. They are capable of simulating extremely large non-uniform domains due to the natural, continuum adaptivity to density changes. In contrast, the mainstream grid-based approaches require the generation and dynamic adaptation of large meshes, a task that still remains challenging at large scales due to algorithmic complexity and load balancing. The adaptive mesh refinement, a necessary feature of any grid-based method dealing with non-uniform problems, often induces artifacts. In addition, the algorithmic complexity of key particle methods insignificantly increases with the increase of spatial dimensions, making a 3D code similar to a 1D code. Particle algorithms are also independent of the geometric complexity of domains. In contrast, there is a huge increase in algorithmic complexity of a 3D mesh generation and dynamic adaptation compared to 1D, especially for multiphase problems, as well as the increase associated with the geometric complexity. This leads to difficulties with load balancing at large scales. The implementation of particle methods will result in smaller and simpler codes that are scalable and easily portable to new architectures. As a general observation, particle-based methods bridge the gap between the continuum and atomistic approaches at refined resolution achievable at exascale.

We have developed a novel meshless method based on Lagrangian particles (LP) capable of transforming the simulation of complex systems at extreme scales [1]. Based on rigorous mathematical approximation theory, our LP method significantly increases the accuracy, convergence order, and robustness of current SPH methods. As a particle method, LP eliminates the need for mesh generation and possess very attractive properties of smooth / continuous adaptivity, improving ideas of the adaptive mesh refinement, and accuracy in resolving complex multiphase boundaries. The LP code is currently being used for the simulation of the plasma jets and liners for the hybrid magneto-inertial fusion concept and high-power accelerator targets.

Particle-based simulation of high power mercury jet target for Muon Accelerator project; dispersion of mercury jet after interaction with 24 GeV, 12 teraproton bunch is shown.

Generalizing the LP ideas to elliptic Vlasov-Poisson-type problems, we have proposed the Adaptive Particle-in-Cloud method (AP-Cloud) [2,3], a highly adaptive replacement for the traditional Particle-in-Cell method (PIC) that eliminates the traditional mesh, replacing it with an octree data structure. The traditional particle-in-cell (PIC) method is not optimal in terms of the balance of errors of the differential operator discretization and source integration; it is also inaccurate when the particle distribution is highly non-uniform. Our method replaces the Cartesian grid in the traditional PIC with adaptive computational nodes or particles, to which the charges from the physical macroparticles are assigned by a weighted least-square approximations. The partial differential equation is then discretized using a generalized finite difference (GFD) method and solved with fast linear solvers. The density of computational particles is chosen adaptively, so that the error from GFD and that from the source integration are balanced and the total error is approximately minimized. The method is independent of geometrical shape of computational domains and free of artificial spurious sources typical for AMR-PIC (that require a special mitigation methods). Results of verification tests using electrostatic problems of particle beams with halo demonstrate that AP-Cloud achieved 30 – 50 times better accuracy compared to PIC.

AP-Cloud Method: distribution of physical particles (left), computational and vacuum particles (middle), and comparison of accuracy with PIC.

Hybrid particle-mesh methods have an edge for problems involving particles and electromagnetic fields. Our third key development, the particle-in-cell electromagnetics code SPACE [4], is a Maxwell equations solver for relativistic particles and fields that implements state-of-art algorithms in computational electrodynamics. The novelty of the code, making it unique compared to a variety of EM codes used in accelerator and plasma science communities, is its ability to resolve atomic transformations and plasma chemistry. The code has been currently used for the support of High Pressure RF Cavity program at Fermilab and for the simulation of processes relevant to eRHIC.

Electromagnetic PIC simulation of relativistic particle beams and electromagnetic fields.

Recent publications

1. R. Samulyak, H. Chen, W. Li, Lagrangian Particle Method for compressible fluid dynamics, J. Comput. Phys., 2015 (submitted). Also in Proc. of SIAM Conf. on Comput. Sci. & Engineering, March 14 – 18, 2015, Salt Lake City, Utah.

2. X. Wang, R. Samulyak, X. Jiao, Optimal solution to classical particle-in-cell problem, Proc. IPAC-2015, paper MOPWA003.

3. X. Wang, R. Samulyak, X. Jiao, Adaptive Particle-in-Cloud: optimized algorithm for Vlasov-Poisson problems, J. Comput. Phys., 2015 (submitted).

4. K. Yu, R. Samulyak, SPACE code for beam-plasma interaction, Proc. IPAC-2015, paper MOPMN012. (JCP paper in preparation).

5. K. Yu, R. Samulyak, M. Chung, A. Tollestrup, K. Yonehara, B. Freemire, Simulation of Beam-Induced Plasma in Gas Filled Cavities, In Proc. IPAC-2015, Paper MOPMN017.

6. J. Ma, R. Samulyak, V. Litvinenko, K. Yu, In Proc. IPAC-2015, May 3-8, 2015, Richmond, VA.Paper MOPMN019.