Physical models

PRISM solves Maxwell's equations in first-order form for the electric displacement $\mathbf{D}$ and the magnetic induction $\mathbf{B}$, in vacuum ($\mathbf{D}={\epsilon }_0\mathbf{E}$, $\mathbf{B}={\mu }_0\mathbf{H}$), driven by a current density $\mathbf{J}$:

$$\frac{\partial \mathbf{D}}{\partial t}=\mathrm{\nabla }\times \frac{\mathbf{B}}{{\mu }_0}-\mathbf{J},\frac{\partial \mathbf{B}}{\partial t}=-\mathrm{\nabla }\times \frac{\mathbf{D}}{{\epsilon }_0},\mathrm{\nabla }\cdot \mathbf{D}=\rho ,\mathrm{\nabla }\cdot \mathbf{B}=0.$$

In conservation form ${\partial }_t\mathbf{q}+\sum _d{\partial }_{x_d}{\mathbf{F}}_d(\mathbf{q})=\mathbf{S}$, with the fluxes of $(\mathbf{D},\mathbf{B})$

$$F_x=(0,\frac{B_z}{{\mu }_0},-\frac{B_y}{{\mu }_0},0,-\frac{D_z}{{\epsilon }_0},\frac{D_y}{{\epsilon }_0}),$$

and $F_y$, $F_z$ by cyclic permutation; the source is $-\mathbf{J}$ on $\mathbf{D}$. The characteristic speeds are $\pm c_0$, $c_0=1/\sqrt{{\epsilon }_0{\mu }_0}$, and two zero speeds. $\mathbf{J}$ has no flux: it is set at every stage by the coils and, under PIC, deposited by the particles.

Models

[physics] physical_model selects one of four models (input; an unknown value is fatal):

Model	physical_model	Current and charge	Units
Electromagnetic	EM (and aliases)	$\mathbf{J}$ from the coils	SI
Non-dimensional electromagnetic	AdimEM	as EM	$L_0$, $B_0$ (below)
Particle-in-cell	PIC	$\mathbf{J}$ from the coils and the particles; $\rho$ from the particles, carried in the state	SI
Non-dimensional PIC	AdimPIC	as PIC	$L_0$, $B_0$

The particles, their deposition and their push are on the particle-in-cell page.

State vector

The state width nv follows from the model and the divergence correction, with one source for both the width and the variable names (physics%var_phi, physics%var_psi, var_Jx; #61):

Slot	Variable	Present
1–3	$D_x,D_y,D_z$	always
4–6	$B_x,B_y,B_z$	always
7	$\phi$ (GLM scalar of $\mathbf{D}$) or $\psi$ (of $\mathbf{B}$)	hyperbolic with D or B
7, 8	$\phi$, $\psi$	hyperbolic with DB
next 3	$J_x,J_y,J_z$	always
last	$\rho$	PIC models

So nv = 9 (EM), 10 (one GLM scalar, or PIC), 11 (DB, or PIC with one scalar) and 12 (PIC with DB), the maximum NV_MAX = 12; a larger nv is fatal at initialization.

Physical constants

adam_prism_parameters:

Constant	Value
${\mu }_0$	1.256637e-6 H/m
${\epsilon }_0$	8.854187e-12 F/m
$c_0$	$1/\sqrt{{\mu }_0{\epsilon }_0}$
electron charge	-1.6e-19 C
electron mass	9.11e-31 kg
ion mass (H⁺)	1.67262192595e-27 kg
neutral mass	ion mass + electron mass
$k_B$	1.380649e-23 J/K

Non-dimensional form

Under AdimEM and AdimPIC the input stays in SI and PRISM scales it by the reference length $L_0$ and field $B_0$ ([physics] L0, B0, both required and positive):

$$T_0=\frac{L_0}{c_0},D_0=B_0{c}_0{\epsilon }_0,I_0=\frac{B_0{L}_0}{{\mu }_0},{\rho }_0=\frac{D_0}{L_0},Q_0={\rho }_0{L}_0^3.$$

The equations become ${\partial }_t\mathbf{D}=\mathrm{\nabla }\times \mathbf{B}-\mathbf{J}$, ${\partial }_t\mathbf{B}=-\mathrm{\nabla }\times \mathbf{D}$, with unit speed of light. Lengths (grid, coil geometry, pulse widths), times, fields and currents are converted at initialization.

Divergence correction

The discrete operators keep $\mathrm{\nabla }\cdot \mathbf{B}$ at round-off on a uniform grid (numerics), but not across a 2:1 AMR face, a deposit that does not conserve charge, or a source that is not divergence-free. [numerics] divergence_correction = hyperbolic adds the generalized-Lagrange-multiplier (GLM) scalars selected by constrained_transport (D: $\phi$, B: $\psi$, DB: both):

$$\frac{\partial \mathbf{D}}{\partial t}=\mathrm{\nabla }\times \frac{\mathbf{B}}{{\mu }_0}-\mathrm{\nabla }\phi -\mathbf{J},\frac{\partial \phi }{\partial t}=-c_h^2(\mathrm{\nabla }\cdot \mathbf{D}-\rho )-\alpha \phi ,$$$$\frac{\partial \mathbf{B}}{\partial t}=-\mathrm{\nabla }\times \frac{\mathbf{D}}{{\epsilon }_0}-\mathrm{\nabla }\psi ,\frac{\partial \psi }{\partial t}=-c_h^2\mathrm{\nabla }\cdot \mathbf{B}-\alpha \psi ,$$

with $\rho$ present only under PIC. The cleaning speed is $c_h=\chi c_0$ ([physics] chi, required, positive) and $\alpha$ the damping rate ([physics] alpha, default 0; in 1/s under EM and PIC, in units of $1/T_0$ under the non-dimensional models, where it is not converted; with $\alpha >0$ the time step is also capped at $1/\alpha$). The fastest signal is then $max(1,\chi )c_0$, which sets the CFL time step.

• hyperbolic without D, B or DB is refused at initialization: there would be no scalar to carry.
• poisson (a projection) is refused: no backend implements it.
• no ignores constrained_transport with a warning.

The full matrix of constrained_transport × divergence_correction is in the input hazards.