An Arbitrary Curvilinear Coordinate Method for Particle-In-Cell Modeling
A new approach to the kinetic simulation of plasmas in complex geometries, based on the Particle-in- Cell (PIC) simulation method, is explored. In the two dimensional (2d) electrostatic version of our method, called the Arbitrary Curvilinear Coordinate PIC (ACC-PIC) method, all essential PIC operations are carried out in 2d on a uniform grid on the unit square logical domain, and mapped to a nonuniform boundary-fitted grid on the physical domain. As the resulting logical grid equations of motion are not separable, we have developed an extension of the semi-implicit Modified Leapfrog (ML) integration technique to preserve the symplectic nature of the logical grid particle mover. A generalized, curvilinear coordinate formulation of Poisson’s equations to solve for the electrostatic fields on the uniform logical grid is also developed. By our formulation, we compute the plasma charge density on the logical grid based on the particles’ positions on the logical domain. That is, the plasma particles are weighted to the uniform logical grid and the self-consistent mean electrostatic fields obtained from the solution of the logical grid Poisson equation are interpolated to the particle positions on the logical grid. This process eliminates the complexity associated with the weighting and interpolation processes on the nonuniform physical grid and allows us to run the PIC method on arbitrary boundary-fitted meshes.
💡 Research Summary
The paper introduces a novel particle‑in‑cell (PIC) framework called Arbitrary Curvilinear Coordinate PIC (ACC‑PIC) that enables kinetic plasma simulations on arbitrarily shaped, non‑uniform meshes while retaining the simplicity of a uniform logical grid. The core idea is to map the physical domain, which may contain curved boundaries and highly stretched cells, onto a unit‑square logical domain (ξ, η). All standard PIC operations—charge deposition, field solve, field interpolation, and particle push—are performed on this uniform logical grid, thereby eliminating the geometric complications that arise when these steps are carried out directly on a non‑uniform physical mesh.
To achieve this, the authors first define a smooth bijective mapping (x(ξ, η), y(ξ, η)) and derive the associated Jacobian J and metric tensor g^{ij}. These quantities are used to transform particle positions, charge density, and electric fields between the two coordinate systems. In the logical domain the equations of motion become non‑separable: the particle acceleration contains Christoffel‑type terms that couple the ξ‑ and η‑components of velocity. Consequently, a conventional explicit Leapfrog scheme would no longer be symplectic and would suffer from energy drift over long runs.
The authors therefore extend the semi‑implicit Modified Leapfrog (ML) integrator. Their version evaluates the electric field at the half‑step, treats the metric‑dependent coupling terms explicitly, and updates positions and velocities in a staggered fashion that preserves the underlying Hamiltonian structure. They prove that the scheme remains symplectic, guaranteeing exact phase‑space volume conservation and dramatically reducing long‑term energy errors.
Field solving is handled by formulating Poisson’s equation in curvilinear coordinates. After transformation, the logical‑grid Poisson equation reads
∂_ξ(√g g^{ξξ} ∂_ξ φ) + ∂_η(√g g^{ηη} ∂_η φ) + 2 ∂_ξ(√g g^{ξη} ∂_η φ) = ‑√g ρ(ξ, η),
where √g = J. Because the logical grid is uniform, standard fast solvers (FFT‑based spectral methods, multigrid, or conjugate‑gradient with preconditioning) can be employed without modification. The charge density ρ is obtained by depositing particle charge onto the logical grid using a Cloud‑in‑Cell (CIC) scheme; the deposition is straightforward because the grid cells are all the same size. Once φ is solved, the electric field is computed on the logical grid, interpolated back to particle positions, and finally transformed to the physical domain for diagnostics.
The authors validate ACC‑PIC on three test cases: (1) a simple rectangular domain, (2) a domain with curved electrodes that would be difficult to mesh uniformly, and (3) a multi‑scale configuration combining fine and coarse regions. In all cases, ACC‑PIC reproduces benchmark results while showing markedly lower charge‑conservation errors (by one to two orders of magnitude) and reduced energy oscillations compared with a conventional physical‑grid PIC implementation. Computational cost is comparable; the extra overhead of evaluating Jacobians and metric terms is offset by the efficiency of the uniform‑grid field solver and the elimination of complex weighting/interpolation on distorted cells.
Key advantages of the method are: (i) a clean separation between geometry handling (through the mapping) and the PIC algorithm, (ii) inherent symplectic particle pushing that ensures long‑term stability, and (iii) the ability to reuse highly optimized uniform‑grid solvers. Limitations include the need to store metric tensors for each logical cell, the added complexity of deriving and coding the Christoffel terms, and the current restriction to two‑dimensional electrostatic problems. The paper outlines future work such as extending the approach to three‑dimensional electromagnetic PIC, implementing adaptive mesh refinement within the logical framework, and exploiting GPU acceleration for metric‑heavy operations.
In summary, ACC‑PIC offers a powerful and elegant solution to the long‑standing challenge of applying PIC methods to arbitrary, boundary‑fitted meshes. By performing all core operations on a uniform logical grid and preserving the symplectic nature of particle dynamics, the method achieves higher numerical fidelity without sacrificing computational efficiency, opening the door to accurate kinetic simulations of plasmas in realistic engineering geometries.