PRISM

PRISM — Plasma Research usIng Simulation Methods — is an electromagnetic and plasma simulation application built on the ADAM framework. It solves the Maxwell equations for computational electromagnetics and, optionally, couples them to charged-particle dynamics via a Particle-in-Cell (PIC) method.

Physical Models

PRISM supports two physical models, selected in the INI file under [physics] physical_model:

Model	INI key	Description
Electromagnetic	electromagnetic	Solves the Maxwell equations for D and B with a prescribed current source J
Particle-in-Cell	PIC	Couples Maxwell equations to individual charged-particle trajectories; current J and charge density ρ are deposited from particle positions

Governing Equations

PRISM solves the first-order hyperbolic form of the Maxwell equations:

$$\frac{\partial \mathbf{q}}{\partial t}+\mathrm{\nabla }\cdot \mathbf{F}(\mathbf{q})=\mathbf{S}$$

The conservative state vector is $\mathbf{q}=(D_x,D_y,D_z,B_x,B_y,B_z,J_x,J_y,J_z{)}^{\mathrm{\top }}$ and the flux components are:

$$F_x=\left(\begin{array}{c}0\\ B_z/\mu \\ -B_y/\mu \\ 0\\ -D_z/\epsilon \\ D_y/\epsilon \\ 0\\ 0\\ 0\end{array}\right)F_y=\left(\begin{array}{c}-B_z/\mu \\ 0\\ B_x/\mu \\ D_z/\epsilon \\ 0\\ -D_x/\epsilon \\ 0\\ 0\\ 0\end{array}\right)F_z=\left(\begin{array}{c}{B}_y/\mu \\ -B_x/\mu \\ 0\\ -D_y/\epsilon \\ D_x/\epsilon \\ 0\\ 0\\ 0\end{array}\right)$$

The wave speeds are $\pm c_0=\pm 1/\sqrt{{\epsilon }_0{\mu }_0}$; the current components $J_{x,y,z}$ have zero flux (source-only variables).

Source Layout

src/app/prism/
├── common/                                      # Shared across all backends
│   ├── adam_prism_parameters.F90                # Physical constants and eigenvector matrices
│   ├── adam_prism_physics_object.F90            # Physical model, variable counts, eigenvectors
│   ├── adam_prism_numerics_object.F90           # Temporal and spatial scheme selection
│   ├── adam_prism_common_object.F90             # prism_common_object base type
│   ├── adam_prism_bc_object.F90                 # Boundary conditions handler
│   ├── adam_prism_ic_object.F90                 # Initial conditions handler
│   ├── adam_prism_rk_bc_object.F90              # RK sub-stepping for boundary conditions
│   ├── adam_prism_io_object.F90                 # I/O handler
│   ├── adam_prism_time_object.F90               # Time integration handler
│   ├── adam_prism_coil_object.F90               # Current-carrying coil sources
│   ├── adam_prism_fWLayer_object.F90            # Far-wave layer (absorbing BC layer)
│   ├── adam_prism_external_fields_object.F90    # Externally prescribed fields
│   ├── adam_prism_pic_object.F90                # Particle-in-Cell configuration
│   ├── adam_prism_particle_injection_object.F90 # Particle injection handler
│   ├── adam_prism_leapfrog_pic_object.F90       # Leapfrog PIC time integrator
│   ├── adam_prism_rk_pic_object.F90             # Runge-Kutta PIC time integrator
│   ├── adam_prism_riemann_library.F90           # Maxwell Riemann solvers (LLF, HLL) and flux routines
│   └── adam_prism_common_library.F90            # Barrel re-export of all common modules
├── cpu/                                         # CPU backend
│   ├── adam_prism_cpu.F90                       # Entry point (program)
│   └── adam_prism_cpu_object.F90                # prism_cpu_object implementation
└── fnl/                                         # OpenACC GPU backend
    ├── adam_prism_fnl.F90                       # Entry point (program)
    ├── adam_prism_fnl_object.F90                # prism_fnl_object implementation
    ├── adam_prism_fnl_kernels.F90               # OpenACC GPU kernels
    ├── adam_prism_fnl_external_fields_kernels.F90 # OpenACC external-fields kernels
    ├── adam_prism_fnl_coil_object.F90           # GPU coil handler
    ├── adam_prism_fnl_fWLayer_object.F90        # GPU far-wave layer handler
    └── adam_prism_fnl_library.F90               # Barrel re-export of all FNL modules

Field Variables

Variable counts are determined at runtime by the physical model and divergence correction settings:

Group	Variables	Count	Notes
Conservative	$D_x,D_y,D_z,B_x,B_y,B_z$	nv_c = 6	Always present
Source	$J_x,J_y,J_z$	nv_s = 3	Always present; zero flux
Divergence cleaning	$\phi$ (D) and/or $\psi$ (B)	nv_cl = 0, 1, 2	Optional hyperbolic cleaning scalars
PIC charge density	$\rho$	nv_pic = 0, 1	Present only in PIC model

Total: nv = nv_c + nv_s + nv_cl + nv_pic

Configuration	nv
EM, no cleaning	9
EM + D or B cleaning	10
EM + D and B cleaning	11
PIC, no cleaning	10
PIC + D or B cleaning	11
PIC + D and B cleaning	12

Maximum supported: NV_MAX = 11.

Numerical Methods

Temporal Schemes

Scheme	INI key	Description
Runge-Kutta	RUNGE_KUTTA	Explicit RK (order configurable)
Leapfrog	LEAPFROG	Staggered-in-time symplectic scheme
Blanes-Moan	BLANES_MOAN	High-order splitting method
Commutator-Free Magnus	COMMUTATOR_FREE_MAGNUS	CFM exponential integrator

Spatial Schemes

Scheme	INI key	Description
WENO	WENO	High-order weighted essentially non-oscillatory reconstruction
FD centered	FD_CENTERED	Finite-difference centered stencil
FV centered	FV_CENTERED	Finite-volume centered stencil

WENO reconstruction operates on either conservative variables (CONSERVATIVE) or characteristic variables (CHARACTERISTICS), selected via [numerics] reconstruction_variables.

Riemann Solvers (adam_prism_riemann_library)

Solver	Description
LLF (Rusanov)	Local Lax-Friedrichs: $F=\frac{1}{2}(F_L+F_R-c_0(q_R-q_L))$
HLL	Harten–Lax–van Leer two-wave solver

Divergence Control

Maxwell's equations require $\mathrm{\nabla }\cdot \mathbf{D}=\rho$ and $\mathrm{\nabla }\cdot \mathbf{B}=0$. PRISM offers three strategies configured via [numerics] divergence_correction:

Strategy	INI key	Description
None	(absent)	No divergence control
Poisson	POISSON	Elliptic cleaning via a Poisson solve
Hyperbolic	HYPERBOLIC	Dedner-type hyperbolic cleaning: augments system with scalar φ/ψ advecting divergence errors at speed $\chi c_0$

Constrained Transport (CT) can additionally be applied to D, B, or both, controlled by [numerics] constrained_transport = D | B | DB | NO.

Source Terms: Coils

The prism_coil_object models current-carrying coils as volumetric current density sources $\mathbf{J}$:

Coil type	Description
rectangular	Rectangular loop coil with user-defined dimensions and normal direction
circular	Circular loop coil

Current type	Description
DC_current	Constant (DC) current
AC_current	Sinusoidal (AC) current with amplitude, frequency, and initial phase

Coil normals are specified as +x, -x, +y, -y, +z, or -z. Coil $\mathbf{J}$ contributions are computed each time step and registered for optional output as auxiliary fields (j_vec_1/2/3, f_Gauss).

Particle-in-Cell (PIC)

When physical_model = PIC, PRISM tracks charged macro-particles alongside the Maxwell field. Each particle carries 8 state variables stored in q_pic(8, particle_number):

Index	Variable	Description
1	x	Position x
2	y	Position y
3	z	Position z
4	vx	Velocity x
5	vy	Velocity y
6	vz	Velocity z
7	charge	Particle charge
8	mass	Particle mass

Field values at particle locations are stored in pic_fields(8, particle_number). Particle time integration uses either Leapfrog (LEAPFROG) or Runge-Kutta (RUNGE_KUTTA), configured independently from the field scheme.

Two problem types are supported:

Type	INI key	Description
Plasma	plasma	Ensemble of charged particles
Single particle	single_particle	Single test-particle trajectory

prism_common_object Overview

The base type aggregates all framework and PRISM sub-objects:

ADAM framework objects: mpih_object, adam_object, field_object, grid_object, amr_object, ib_object, slices_object, leapfrog_object, blanesmoan_object, cfm_object, rk_object, weno_object, flail_object

PRISM sub-objects: prism_physics_object, prism_numerics_object, prism_io_object, prism_bc_object, prism_ic_object, prism_rk_bc_object, prism_time_object, prism_coil_object, prism_fWLayer_object, prism_external_fields_object, prism_pic_object, prism_particle_injection_object, prism_leapfrog_pic_object, prism_rk_pic_object

Field arrays (all shape (nv, ni, nj, nk, nb)):

Array	Description
q	Conservative field: D, B, J (+ optional cleaning scalars and ρ)
dq	Residuals RHS
curl	Curl of field components (optional output)
divergence	Divergence of field components (optional output)

PIC arrays: q_pic(8, particle_number), pic_fields(8, particle_number)

Energy diagnostics: energy_D(:), energy_B(:) (time histories), rms_energy_error_D/B

Backends

Backend	Entry point	Key type	Accelerator
CPU	adam_prism_cpu.F90	prism_cpu_object	MPI + OpenMP
FNL	adam_prism_fnl.F90	prism_fnl_object	MPI + OpenACC (NVIDIA GPU)

The FNL backend adds GPU companion objects (field_fnl_object, rk_fnl_object, weno_fnl_object, etc.) and device-side pointer arrays (q_gpu, dq_gpu, flx_f_gpu, curl_gpu, …). The CPU backend implements all differential operators (curl, divergence, gradient, Laplacian, derivatives 1–4) via procedure pointers, dispatched at initialisation based on scheme_space.

Building

# CPU backend (GNU compiler)
FoBiS.py build -mode prism-gnu

# CPU debug build
FoBiS.py build -mode prism-gnu-debug

# FNL (OpenACC) backend
FoBiS.py build -mode prism-fnl-nvf-oac

Executables are written to exe/adam_prism_cpu and exe/adam_prism_fnl.

Running

mpirun -np <N> exe/adam_prism_cpu [input.ini]
mpirun -np <N> exe/adam_prism_fnl [input.ini]

If no argument is given, input.ini in the working directory is used.

License

PRISM is part of the ADAM framework, released under the GNU Lesser General Public License v3.0 (LGPLv3).

Copyright (C) Andrea Di Mascio, Federico Negro, Giacomo Rossi, Francesco Salvadore, Stefano Zaghi.