1. An electromagnetic pulse in a periodic box

The first PRISM tutorial runs a Gaussian pulse once around a periodic box: the build, the input, the energy history, and how far the pulse comes back to where it started.

Build

fobis build --mode prism-cpu-gnu         # exe/adam_prism_cpu; links FFTW (the t0 elliptic solves)

The input

src/tests/prism/regression/periodic-pulse-blocks/input.ini, the keys that define the problem (the full reference is PRISM input):

[physics]
physical_model = EM              ; Maxwell's equations in D and B, SI units

[numerics]
scheme_space          = fd_centered
scheme_time           = runge_kutta
constrained_transport = no
divergence_correction = no

[fdv]
fdv_scheme = fd
fdv_order  = 6                   ; 6th-order centred derivatives

[runge_kutta]
scheme = runge-kutta-ssp-54

[grid]
ni     = 12                      ; 12^3 cells per block
emin_x = -0.1                    ; the box [-0.1, 0.1]^3 (metres)
emax_x = 0.1

[amr]
iu_ref_levels = 2                ; 64 blocks: 48^3 cells, dx = 0.2/48

[initial_conditions]
type            = gaussian_pulse
pulse_direction = +x
r0              = 0.07           ; centre of the packet along x
sigma           = 0.015          ; width
k0              = 0.0            ; no carrier: a pure Gaussian
B0              = 10.0

[bc_x_min]
type = periodic                  ; all six faces periodic

[time]
it_max   = 192
CFL      = 0.5

The pulse is $E_y=c_0{B}_0{e}^{-(x-r_0{)}^2/2{\sigma }^2}$, $B_z=E_y/c_0$, travelling along $+x$ (initial conditions). The time step is $\mathrm{\Delta }t=\mathrm{\Delta }x/(4c_0)$, so 192 steps are exactly one crossing of the box: the pulse leaves through $+x$, comes back through $-x$ and returns to its start.

Run it

mkdir -p run/pulse && cp src/tests/prism/regression/periodic-pulse-blocks/input.ini run/pulse/
cd run/pulse && mpirun -np 2 ../../exe/adam_prism_cpu input.ini > log.txt

It writes the fields every 48 steps (periodic_pulse_blocks-000000000.xdmf … -000000192.xdmf), the restart files, -residuals.dat, -divergence_history.dat and -energy_history.dat.

What to look at

Energy (periodic_pulse_blocks-energy_history.dat, every 10 steps; electric and magnetic energy in joules):

+10  +32 +0.34746257462833344E-010 +0.42220747103426816E+005 +0.42220747103426831E+005 ...
+192 +32 +0.66712814328639916E-009 +0.42220694691084245E+005 +0.42220694691084274E+005 ...

The two energies stay equal (a travelling wave) and their sum drifts by $1.2\cdot {10}^{-6}$ relative over the period: the centred scheme is non-dissipative, and the drift is the Runge–Kutta's.

Divergence: the last row of -divergence_history.dat is zero for $\mathrm{\nabla }\cdot \mathbf{D}$ and $\mathrm{\nabla }\cdot \mathbf{B}$: on a uniform grid the centred divergence of the centred curl vanishes identically (numerics).

The return. src/tests/tutorials/compare_saves.py compares two saves of a run, block by block (it needs h5py: the venv of the first run):

../../exe/.regression-venv/bin/python ../../src/tests/tutorials/compare_saves.py . periodic_pulse_blocks 0 192

Bz       max|q(192) - q(0)| / max|q(0)| = 3.961e-02
Dy       max|q(192) - q(0)| / max|q(0)| = 3.961e-02

A 4% difference after one crossing, and most of it is not the scheme: the initial pulse is set as a Gaussian about r0 = 0.07 that is not wrapped across the periodic faces, so at the face $x=0.1$, $2\sigma$ from its centre, the field jumps by 18% of the peak. That jump travels with the pulse and dominates the return error; a 1-D model of this run gives 4.0% with the cut and 0.14% with a wrapped pulse, and refining the grid does not remove it. Centre the pulse in the box (r0 = 0.0, as the next page does) to see the scheme's own error.

What you learned

• PRISM's input names the model, the space and time schemes, the grid, the initial field and the faces.
• The energy and divergence histories are the first checks: energy flat, divergence zero on a uniform grid.
• Dispersion, not dissipation, is the error of a centred scheme on a narrow pulse.

Next: 2. Reflecting and absorbing faces.