Particle-Grid Characteristic Mapping

• Particle-Grid Characteristic Mapping is a framework that couples lagrangian particles with eulerian grids using invertible characteristic flow maps.
• It employs forward and backward maps with Jacobian tracking and submap decomposition to preserve geometric and physical insights under large deformations.
• Applications in fluid dynamics, plasma, and CFD-DEM show enhanced accuracy and efficiency through exact moment preservation and robust, conservative transfers.

The Particle-Grid Characteristic-Mapping Framework is a family of methodologies for coupling Lagrangian particle representations with Eulerian grids through mathematically precise transport maps, enabling robust, high-fidelity, and conservative transfer of density, momentum, and other quantities between particle and grid descriptions. The framework arises in diverse contexts—fluid and plasma simulation, level-set interface tracking, deep learning for materials, and multiphase CFD—with the unifying principle that particle trajectories encode characteristic flow maps, which can be used to construct invertible mappings between particle and grid (or parameter domain) representations. Characteristic-mapping maintains geometric and physical accuracy over long times and under large deformations, often outperforming direct grid-based or traditional kernel-based schemes.

1. Mathematical Foundations of Characteristic Mapping

The core of the framework is the use of characteristic flow maps, either in physical space or parameter domains, to transport fields or distributions between particles and grids or to reparametrize marker distributions.

Consider a time-dependent velocity field $u(x,t)$ on a domain $\Omega$. The forward flow map

$\varphi(X, t) = x,$

is the solution of

$$\frac{d}{dt} \varphi(X, t) = u(\varphi(X, t), t), \qquad \varphi(X, 0) = X.$$

The backward map $\psi(x, t)$ satisfies $\psi(\varphi(X, t), t) = X$. The evolution of geometric quantities such as Jacobians is given by

$$\frac{DF}{Dt} = \nabla u \cdot F,\qquad \frac{DT}{Dt} = -T \cdot \nabla u,$$

with $F(x, t) = \partial \varphi(x, t) / \partial x$ and $T(x, t) = \partial \psi(x, t)/\partial x$ (Zhou et al., 2024).

Characteristic-mapping in parameter domains leverages backward maps $\Phi_B(u, t)$ defined by

$$\partial_t \Phi_B(u, t) + (w(u, t)\cdot \nabla_u) \Phi_B(u, t) = 0,$$

where $w$ is a redistribution velocity, e.g., generated by a diffusion process, which uniformly redistributes markers on evolving surfaces (Yin et al., 2020).

2. Frameworks and Algorithms for Particle–Grid Characteristic-Mapping

Various implementations exist depending on the application domain:

A. Fluid and Plasma Dynamics

• Particle Flow Map (PFM) framework: Lagrangian particles store long-range and short-range Jacobians of the backward map; these are updated by integrating ODEs along trajectories and used to reconstruct transported quantities and their gradients. An impulse-based Poisson solver on the Eulerian grid enforces incompressibility (Zhou et al., 2024, Li et al., 31 Jan 2026).
• Impulse-form Navier–Stokes: The impulse $m = u + \nabla\phi$ is a gauge-transformed velocity satisfying a characteristic law even for incompressible Navier–Stokes. The impulse is decomposed into a purely advected (geometric) part and a path-integral for viscosity and body-forces along each particle trajectory, enabling exact geometric transport and efficient pressure projection (Li et al., 31 Jan 2026).

B. Surface and Curve Management

• Diffusion-driven redistribution: For explicit evolution of parameterized surfaces $X:U\times[0,T]\to\Omega$, marker sampling becomes nonuniform as the area element $J(u, t)$ distorts. By interpreting $J(u, t)/\int_U J(u, t)du$ as a probability density $\rho(u, t)$, diffusion and the induced transport velocity $w(u,t)=-\nabla\ln\rho$ yield a deformation map $\Phi_B$. Pre-composing $X$ with $\Phi_B$ regularizes marker distributions. Semigroup property of the heat equation allows submap decomposition for efficient reparametrization (Yin et al., 2020).

C. Particle-In-Cell (PIC) in Complex Geometries

• Curvilinear PIC: All particle-field operations are performed on a uniform logical grid $(\xi, \eta)$ mapped to the physical domain via a known coordinate transformation. The symplectic Hamiltonian pusher and Poisson solver account for metric and Jacobian factors; charge deposition and field interpolation remain standard due to uniformity in the logical domain (Fichtl et al., 2012).

D. High-Order Mapping in Velocity Space

• Pseudo-inverse mapping: Grid-to-particle and particle-to-grid mappings are formalized via projections onto finite-element spaces. The use of the Moore–Penrose pseudoinverse ensures exact preservation of polynomial moments up to the FE order, including density, momentum, and energy (for $p\ge2$ elements), yielding significant improvements over bilinear mapping in kinetic plasma PIC codes (Mollén et al., 2020).

3. Workflow and Submap Decomposition

A generic particle–grid characteristic-mapping algorithm involves:

• Advecting particles along the velocity field, integrating the flow map and, where required, its Jacobians.
• Carrying, updating, and manipulating per-particle stashes of transported quantities and their derivatives (impulse, gradients, or higher derivatives).
• Transferring particle values to the Eulerian grid, typically using B-spline or APIC kernels for locally affine accuracy.
• Solving grid-based field equations (e.g., pressure Poisson equation, surface PDEs, or logical-domain Poisson in PIC).
• Interpolating updated grid values back to particles to advance the next step.
• Periodic reinitialization for long-range and short-range maps to control distortion, and, in surface management, for reparameterizing markers after severe deformation (Yin et al., 2020, Zhou et al., 2024).

The semigroup property for diffusion and flow map composition allows for submap decomposition—global maps are formed by sequential application of short-interval maps, which improves efficiency and mitigates numerical error (Yin et al., 2020).

4. Applications and Numerical Performance

Fluid Simulation and Vortex Dynamics

• PFM achieves vorticity preservation in long-time vortex interactions and turbulent flows, maintaining Frobenius-norm error $\|F\cdot T-I\|<10^{-5}$ over 200 steps and less than 1% energy drift—outperforming APIC and Neural Flow Map methods by order-of-magnitude factors in both accuracy and computational efficiency. PFM runs up to $49\times$ faster with $29$–$41\%$ less memory than NFM (Zhou et al., 2024).

Surface and Curve Sampling

• For chaotic 3D curve evolution and complex surface advection, CM-based redistribution maintains uniform sampling (area density standard deviation reduced from $O(1)$ to $O(10^{-2})$). The method supports robust marker management under large deformations, preserving the desired statistical properties of marker distributions on surfaces (Yin et al., 2020).

Level Set Interface Tracking

• PFM-LS maintains volume and sharp interface features under extreme deformation. In 2D Zalesak and LeVeque benchmarks, PFM-LS achieved relative volume errors of $10^{-7}$–$10^{-8}$; in 3D, $99.68\%$–$99.99\%$ volume retention at maximum deformation—significantly better than GARM-LS and BFECC+RM (He et al., 14 Jan 2026).

Deep Learning for Materials Discovery

• ParticleGrid maps molecular structures onto dense $C\times N_x\times N_y\times N_z$ voxel grids using Gaussian-smeared particle contributions, enabling accurate (0.006 MSE) neural property prediction at thousandfold speedup over DFT, with throughput of $\sim5000$ molecules/sec grid generation and $\sim1.2$ms/molecule inference (Zaman et al., 2022).

CFD-DEM Multiphase Flows

• A two-step mapping—coarse graining particles onto multi-layer Fibonacci point clouds followed by projection onto the CFD grid—achieves smooth, stable, and grid-independent coupling from dilute to dense regimes. Convergence surpasses first order with low grid/particle size ratio, and benchmark tests (sedimentation, Ergun columns, fluidized beds, granular collapse) confirm superior accuracy and stability to kernel-based or direct PCM/DPVM approaches (Liu et al., 11 Jun 2025).

Plasma PIC with Curvilinear Grids

• All deposition, field solve, and interpolation are performed on a uniform logical grid; metric coefficients enter only in the field-pusher and Poisson solve. The method preserves charge and phase-space volume, and supports simulations in arbitrarily complex, boundary-fitted domains (Fichtl et al., 2012).

Velocity Space Mapping

• The pseudo-inverse mapping enables exact moment preservation (up to kinetic energy) in velocity space, reducing cumulative errors to machine-precision per time step in neoclassical tokamak PIC runs, eliminating drift seen with bilinear and first-order schemes (Mollén et al., 2020).

5. Complexity, Implementation, and Limitations

Method/Domain	Main Mapping Strategy	Complexity (Per Step)
PFM (fluid)	Dual-scale Jacobians, APIC	$O(N_\text{grid} \log N_\text{grid})$ (Poisson), $O(N_\text{particles})$ (advection)
Curvilinear PIC	Uniform logical grid, canonical Hamiltonian	Particle push and field solve $O(N_p)$, $O(N_g \log N_g)$
Pseudo-inverse velocity	FE projection, sparse matrices	$O(N_p n_b)$ (forward), $O(N_g^3)$ (local inversion)
Diffusion-driven CM	Diffusion, Hermite interp.	$O(G)$ (heat step), $O(M)$ (marker sampling)
Two-step CFD-DEM	Point-cloud coarse graining	$O(N_{pt} N_p)$ (where $N_{pt}$: points/particle)
ParticleGrid for ML	Gaussian smearing, SIMD	$O(C N^3 N_p)$ (grid gen), parallelizable

Implementation involves per-particle storage of positions, multiple Jacobians, and transported quantities (impulse, gradients, buffers). Interpolation uses APIC or B-spline kernels. In high-order mapping, mass-matrix assembly and inversion may be bottlenecks, but are mitigated by small block sizes and high sparsity.

Limitations include increased particle memory for high-order or multi-scale quantities, particle under-sampling in extreme deformations (necessitating adaptive reseeding), and, in two-step CFD-DEM, restriction to spherical shapes unless point clouds are generalized. Large 3D interface problems may become bottlenecked by per-particle memory, requiring compression or hybrid architectures (Yin et al., 2020, He et al., 14 Jan 2026, Liu et al., 11 Jun 2025).

6. Extensions and Outlook

Numerous directions extend the applicability and generality of the Particle–Grid Characteristic-Mapping Framework:

• Higher dimensions and anisotropic redistribution: The formalism applies unchanged to domains $U\subset\mathbb{R}^d$ and supports general diffusion or Fokker–Planck operators for feature-adaptive sampling (Yin et al., 2020).
• Integration with Eulerian and Lagrangian solvers: The framework is amenable to insertion into finite element/finite volume solvers for surface PDEs, hybrid Eulerian–Lagrangian schemes, or deep learning pipelines (Yin et al., 2020, Zaman et al., 2022, Zhou et al., 2024).
• Non-spherical and deformable particles: Extensions to arbitrary surfaces for CFD-DEM or interface problems are enabled by generalized point cloud constructions (Liu et al., 11 Jun 2025).
• GPU parallelization and scalability: All reported schemes admit data-parallel particle and grid loops, efficient SIMD implementations, and integration with modern accelerator hardware (Zaman et al., 2022, He et al., 14 Jan 2026).
• Advanced moment preservation: Pseudo-inverse velocity mapping lays a rigorous foundation for higher-order conservation in PIC contexts, eliminating cumulative drifts and aligning with structure-preserving methodologies (Mollén et al., 2020).

The Particle–Grid Characteristic-Mapping Framework unifies a broad class of multi-physics and data-driven methods, providing a robust, extensible toolkit for accurate, conservative, and geometry-aware coupling of Lagrangian and Eulerian descriptions across computational science and machine learning domains.