The n = 1 polytrope: equilibrium and collapse

ID	Group	Milestone	Backends	Figure	Regression
SV-7, SV-8, SV-14 to SV-17	Self-gravity	M8 (#81), P4; #87, P3, P4; #88; #91	CPU, FNL	yes	selfgravity-star, selfgravity-star-amr

Case and scope

An n = 1 polytrope star under the isolated self-gravity of SV-6, centred in a closed box. The n = 1 polytrope is the Lane–Emden equilibrium $p=K{\rho }^2$ with a closed-form density, so the discrete scheme holds it up to its truncation error, and halving its pressure makes it collapse, bounce and ring.

What the case catches:

• SV-7, the equilibrium: a potential or a force inconsistent with the pressure gradient. The spurious motion of the discrete scheme around the continuum equilibrium (the drift of the central density, the Mach number inside the star) must shrink with the grid;
• SV-8, the collapse: a leak in the exact exchange of $E$ and $W$ through the isolated solver. Its symmetry (an even kernel) must give the same exactness as the periodic solver: $E+W$ drifts only at the Runge–Kutta order while $E$ and $W$ exchange, and the momentum stays at round-off;
• SV-14, the star on AMR (#87 P3): isolated self-gravity by the multigrid's symmetric James step (SV-13) coupled to the scheme. The equilibrium on a grid with the star refined must have less spurious motion than on the uniform base; the collapse must keep the exact exchange (the per-stage reflux carries the seam gravity work); a regrid that refines a boundary block must stop the run (D-87-4).

Models and numerics

Item	Choice	Code
Equations	compressible Euler, $\gamma =2$ (the adiabatic exponent of $p=K{\rho }^2$), isolated self-gravity, $G=1$	models
Poisson solver	FFT with isolated boundaries (poisson_bc = isolated): zero padding and the lattice Green's function; on AMR (SV-14) the multigrid's James step, its coarsest level on the same FFT solver	adam_poisson_mg_object.F90, adam_poisson_fft_object.F90, adam_fnl_poisson_fft_object.F90
Coupling	one solve per Runge–Kutta stage, the potential path	solve_potential, adam_flume_common_object.F90; add_potential_sources, adam_flume_cpu_object.F90
Space	WENO5 (weno-u-5) flux splitting on characteristic variables (weno)	numerics
Time	SSP Runge–Kutta, 3 stages, CFL 0.4 (SV-8: 0.4, 0.2, 0.1)	adam_rk_object.F90
Positivity limiter	SV-7: cell (its gravity energy correction, M7-G1 option B); SV-8: none	limit_positivity, adam_flume_cpu_object.F90; numerics
Boundaries	wall-inviscid on every face: a closed box, no mass or energy crosses it, so $E+W$ has no boundary term	boundary conditions; adam_flume_bc_object.F90
Initial condition	polytrope: the star and its atmosphere	set_initial_conditions, adam_flume_ic_object.F90

Initial and boundary conditions

Item	Value
Star	polytrope: n = 1, radius $a=1/2$, ${\rho }_c=1$, $G=1$, $p=K{\rho }^2$ with $K=2G{a}^2/\pi$ (the Lane–Emden equilibrium), $\gamma =2$; mass $4{\rho }_c{a}^3/\pi$, mean density $3{\rho }_c/{\pi }^2$, free-fall time of the mean density $t_{ff}=\sqrt{3\pi /(32G\overline{\rho })}=0.98$
Atmosphere	isothermal at the central temperature ($p_1=K{\rho }_c{\rho }_1$) and hydrostatic in the star's field, $\rho ={\rho }_1{e}^{GM/T(1/r-1/a)}$ outside; inside, ${\rho }_1$ and $p_1$ added to the star
Box	$[-1,1{]}^3$, an octree, uniform; 8 cells per block side, 4 ghosts; wall-inviscid on every face
Ranks	2
SV-7, equilibrium (star)	star_pressure_factor 1, atmosphere ${\rho }_1={10}^{-4}{\rho }_c$, positivity limiter cell; ${32}^3$ and ${64}^3$ (levels 2 and 3) to $t=2$, CFL 0.4, a checkpoint every 10 steps
SV-8, collapse (starc)	star_pressure_factor 0.5, atmosphere ${\rho }_1={10}^{-2}{\rho }_c$, no limiter; ${32}^3$ to $t=1.5$ at CFL 0.4, 0.2, 0.1
SV-14, the star on AMR	a smaller star, radius $a=0.3$, so that the 2:1 seam of the box $[-1/2,1/2{]}^3$ lies in the atmosphere, 0.2 off the surface (with $a=1/2$ the seam cuts the surface at the face centres and the light atmosphere goes non-finite, #88); the box refined once over the ${32}^3$ base (--box: 120 blocks, the star at the ${64}^3$ spacing, 9.6 cells per radius)
SV-14, equilibrium	as SV-7 (atmosphere ${10}^{-4}{\rho }_c$, limiter cell, to $t=2$), on the AMR grid and on the uniform ${32}^3$ grid
SV-14, collapse	as SV-8 (atmosphere ${10}^{-2}{\rho }_c$, no limiter, to $t=1.5$, CFL 0.4, 0.2, 0.1) on the AMR grid with reflux_cadence = stage
SV-14, refusal	starc --regrid 2 --boundary-box: a runtime marker on $[1/2,1{]}^3$, at the boundary; the first regrid refines a boundary block
SV-15, runtime-AMR collapse	starc --radius 0.3 --reflux-cadence stage --regrid 5: the SV-14 star on the uniform ${32}^3$ base (4.8 cells per radius), atmosphere ${10}^{-2}{\rho }_c$, no limiter, to $t=1.5$; a regrid every 5 steps by a Löhner marker on the density (one level more, the atmosphere's density as its floor), the first one refining the 8 inner blocks; regrid_prolongation = conservative-state, and the default conservative reported
SV-17, dual energy	the SV-15 case (starc --radius 0.3 --reflux-cadence stage --regrid 5, the default prolongation, no Jeans marker) with --physical-model euler-de, the dual-energy switch at its default 0.05 (DE)
SV-16, Jeans refinement	starc --radius 0.22 --reflux-cadence stage --regrid 5 --jeans 8: a smaller star (3.5 cells per radius on the base; its Jeans length is $\sqrt{2}a$, 5 base cells, since ${\lambda }_J^2=\pi \gamma p/(G{\rho }^2)=\pi K/G=2a^2$ for the collapsing $\gamma =2$ polytrope, uniform inside it and unchanged by an adiabatic collapse), the SV-15 Löhner marker and a Jeans marker of 8 cells per Jeans length, the initial AMR on (amr_iterations = 1); the default prolongation; the same star without the Jeans marker reported
Regression selfgravity-star	star --levels 1: ${16}^3$ (2×2×2 blocks of $8^3$), atmosphere ${10}^{-2}{\rho }_c$, no limiter, 20 steps
Regression selfgravity-star-amr	starc --radius 0.3 --box --reflux-cadence stage: SV-14's collapse, 120 blocks of $8^3$, 20 steps

Input files

File	Role
src/tests/flume/verification/selfgravity/make_polytrope.py	writes star and starc (--cells, --levels, --cfl, --time-max, --it-save, --atmosphere, --limiter; on AMR --radius, --box, --regrid, --boundary-box, --reflux-cadence)
src/tests/flume/verification/selfgravity/star.sh	runs the legs sv7, sv8, sv14, sv15, sv16, sv17 (--np N, --legs LIST, --keep; FLUME_EXE selects the backend)
src/tests/flume/verification/selfgravity/star_oracle.py	convergence (SV-7): the drifts on two grids and their reduction
src/tests/flume/verification/selfgravity/jeans_oracle.py	conservation (SV-8): $E+W$ against $\mathrm{\Delta }t$, the momentum
src/tests/flume/regression/selfgravity-star/input.ini	the frozen equilibrium at ${16}^3$, 20 steps
src/tests/flume/regression/selfgravity-star-amr/input.ini	the frozen SV-14 collapse on AMR, 20 steps (#87 P4)
src/tests/flume/verification/make_doc_runs.sh, group isolated	sv7-l{2,3}, sv8-collapse-{0.4,0.2,0.1} (a checkpoint every 10 steps); needs exe/adam_flume_cpu_fft and exe/test_poisson_fft
src/tests/flume/verification/make_doc_runs.sh, group isolated	also sv14-{uniform,box}, sv14-collapse-{0.4,0.2,0.1}
src/tests/flume/verification/make_doc_runs.sh, group collapse-regrid	sv15-{conservative-state,conservative}, sv16-{jeans,control} and sv17-de, a checkpoint every 2 steps (the conservative and control runs stop by design)
src/tests/flume/verification/make_doc_figures.py, fig_selfgravity_star, fig_selfgravity_star_amr, fig_selfgravity_star_regrid	draws the figures (--only selfgravity-star selfgravity-star-amr selfgravity-star-regrid)

cd src/tests/flume/verification/selfgravity && ./star.sh                   # CPU (exe/adam_flume_cpu_fft)
FLUME_EXE=$PWD/../../../../../exe/adam_flume_fnl_fft ./star.sh               # FNL

Results

Top: at ${32}^3$ and ${64}^3$ the core first expands (the central density falls, by 10 % and 4.5 %) and then recompresses past its initial value within two time units; on the finer grid the swing is smaller and the interior Mach number about half. Bottom left: in the collapse $E$ and $W$ exchange 74 % of $|E+W|$ (their curves coincide) while $E+W$ drifts at the Runge–Kutta order, a factor 8 per halving of the CFL number. Bottom right: the collapsing star at its densest checkpoint, $t=1.03$, its central density four times the initial one.

The central density is the mean over the cells within one cell of the centre; the Mach numbers are taken inside $0.8a$ and over the whole star (the cells denser than a tenth of the central density).

Leg	Asserted	Bound	CPU	FNL
SV-7	the largest central-density drift to $t=2$: $9.91\cdot {10}^{-2}$ (${32}^3$), $5.71\cdot {10}^{-2}$ (${64}^3$); reduction	$\ge 1.5$	1.74	1.73
SV-7	the largest Mach number inside $0.8a$: 0.154, 0.075; reduction	$\ge 1.5$	2.05	2.06
SV-7	the largest Mach number in the whole star (set at its surface): 0.43, 0.29; reduction	reported	1.50	1.49
SV-8	drift of $E+W$ at CFL 0.4, 0.2, 0.1: $5.97\cdot {10}^{-7}$, $8.29\cdot {10}^{-8}$, $9.71\cdot {10}^{-9}$; order in $\mathrm{\Delta }t$	$\ge 2.5$	2.85, 3.09	2.85, 3.09 (the same drifts)
SV-8	the largest $|\int \rho \mathbf{u}|$	${10}^{-14}$	$7.3\cdot {10}^{-18}$	$6.9\cdot {10}^{-18}$

SV-14, the star on AMR (#87 P3):

Top: the star of radius 0.3 on the uniform ${32}^3$ grid (4.8 cells per radius) and with the box refined (9.6): on the AMR grid the core swings half as far and the interior Mach number stays at half. Bottom left: the collapse on the AMR grid, $E+W$ at the Runge–Kutta order while $E$ and $W$ exchange 81 % of $|E+W|$ (their curves coincide). Bottom right: the equilibrium star at $t=2$ on its composite grid, the radius dashed; the 2:1 seam lies in the atmosphere.

Leg	Asserted	Bound	CPU	FNL
SV-14	the largest central-density drift to $t=2$: $1.65\cdot {10}^{-1}$ (uniform ${32}^3$), $8.24\cdot {10}^{-2}$ (AMR); reduction	$\ge 1.5$	2.00	2.00
SV-14	the largest Mach number inside $0.8a$: 0.414, 0.189; reduction	$\ge 1.5$	2.19	2.19
SV-14	the largest Mach number in the whole star: 15.2 (uniform: the surface spans 5 cells), 0.36; reduction	reported	42	42
SV-14	collapse on AMR, drift of $E+W$ at CFL 0.4, 0.2, 0.1: $4.76\cdot {10}^{-7}$, $6.07\cdot {10}^{-8}$, $7.72\cdot {10}^{-9}$; order in $\mathrm{\Delta }t$	$\ge 2.5$	2.97, 2.97	2.97, 2.97 (the same drifts)
SV-14	the largest $|\int \rho \mathbf{u}|$; mass drift	${10}^{-14}$; ${10}^{-14}$	$2.2\cdot {10}^{-19}$; $2.4\cdot {10}^{-16}$	$4.3\cdot {10}^{-19}$; $2.4\cdot {10}^{-16}$
SV-14	a regrid that refines a boundary block	refused, exit 1	refused	refused

SV-15 to SV-17, the runtime-AMR collapse (#88, #91):

Top row, SV-15: the SV-14 star collapsing on the uniform ${32}^3$ base, a regrid every 5 steps. Left: the regrid at step 5 refines the 8 inner blocks and lowers the smallest $e/\rho$ from $7.0\cdot {10}^{-3}$ to $5.0\cdot {10}^{-3}$ with either prolongation. With the default conservative the coldest cells then keep cooling while an infall reaches Mach 10: $3.5\cdot {10}^{-4}$ by step 62, non-finite at step 64 ($t=0.70$). With conservative-state the smallest $e/\rho$ never falls below $5.07\cdot {10}^{-3}$, the Mach number stays below 1.8 and the run reaches $t=1.5$. Right: the mass at round-off in both; $E+W$ jumps at the regrid (the potential is solved on the new grid), by $1.5\cdot {10}^{-3}$ with conservative and $3.5\cdot {10}^{-5}$ with conservative-state, and drifts at the time integration's error between regrids. Bottom row, SV-16: the smaller star refined only by the Löhner marker at step 5 cools to non-finite at step 27; with the Jeans marker the initial AMR refines it before the collapse (120 blocks, 64 on the base), no regrid changes the grid, the coldest cell recovers and the run reaches $t=1.5$ with the default prolongation, $E+W$ drifting by $7.6\cdot {10}^{-7}$ with no regrid jump. Bottom row, SV-17: the SV-15 star with euler (the red curve of the top row) and with euler-de: the floor lifts the coldest cells to their adiabat in 6 steps, adding $4.1\cdot {10}^{-7}$ to the total energy, and the run reaches $t=1.5$ with the default prolongation.

Leg	Asserted	Bound	CPU	FNL
SV-15	conservative-state: reaches $t=1.5$, with a regrid that changes the grid	completes	$t=1.5$, 1 regrid changes the grid	$t=1.5$, 1 regrid changes the grid
SV-15	conservative-state: mass and momentum drifts	${10}^{-13}$	mass $1.2\cdot {10}^{-16}$, momentum $3.8\cdot {10}^{-18}$	mass $1.2\cdot {10}^{-16}$, momentum $7.6\cdot {10}^{-18}$
SV-15	conservative, the default	reported	non-finite at step 64	non-finite at step 65
SV-16	Jeans marker: the initial grid refined (blocks at step 0 above the 64 of the base); reaches $t=1.5$	completes	120 blocks at step 0; $t=1.5$	120 blocks at step 0; $t=1.5$
SV-16	Jeans marker: mass and momentum drifts	${10}^{-13}$	mass $2.1\cdot {10}^{-16}$, momentum $1.7\cdot {10}^{-18}$	mass $2.1\cdot {10}^{-16}$, momentum $3.3\cdot {10}^{-18}$
SV-17	euler-de: reaches $t=1.5$	completes	$t=1.5$	$t=1.5$
SV-17	euler-de: mass and momentum drifts	${10}^{-13}$	mass $1.2\cdot {10}^{-16}$, momentum $4.8\cdot {10}^{-19}$	mass $1.2\cdot {10}^{-16}$, momentum $9.6\cdot {10}^{-19}$
SV-17	the energy the entropy floor added	reported	$4.1\cdot {10}^{-7}$ in 6 steps	$4.1\cdot {10}^{-7}$ in 6 steps
SV-16	without the Jeans marker	reported	non-finite at step 27	non-finite at step 33

Related cases: the periodic analogue of the collapse is SV-4; the solver is SV-6, on AMR SV-13; the limiter's gravity energy correction comes from the force (M7-G1).

Findings

• Only the time integration drifts. In the collapse $E$ and $W$ each change by 74 % of $|E+W|$, while $E+W$ drifts by $6\cdot {10}^{-7}$ at CFL 0.4 and falls as $\mathrm{\Delta }{t}^3$: the isolated solver conserves as exactly as the periodic one (SV-4), through its symmetry.
• The positivity limiter breaks the exchange where it acts. Its gravity energy correction (M7-G1, option B) is not part of the flux form. A star in a ${10}^{-4}$ atmosphere needs it at its surface (without it the gas there reaches $|\mathbf{u}|\approx 0.4$ in ten steps and its pressure goes negative), and with it $E+W$ drifts by $2.8\cdot {10}^{-3}$ by $t=4$ at ${32}^3$, against $3\cdot {10}^{-9}$ without it in a ${10}^{-2}$ atmosphere. SV-8 therefore runs in the heavier atmosphere, without the limiter.
• A heavy atmosphere is not an equilibrium test. The ${10}^{-2}$ atmosphere holds 22 % of the star's mass, and the star then pulsates with 13 % of its central density at ${32}^3$; SV-7 uses the light one.
• The equilibrium converges at about first order, set by the star's surface. The n = 1 surface has a kink in the density (${\rho }_s\propto a-r$, $p\propto (a-r{)}^2$), the light atmosphere needs the limiter there (first order where it acts), and the surface is where the largest Mach number sits ($r=0.454$, $a=0.5$): it shrinks only 1.5-fold from ${32}^3$ to ${64}^3$. The pulsation it drives sets the core's drift, which shrinks 1.7-fold, and the interior motion 2.1-fold. A star of radius 8 cells pulsates by 10 % of its central density, of 16 cells by 6 %.
• The bound asserts convergence, not an order. P0 asked for 100 free-fall times; the runs reach 2 (${64}^3$ took 37 min on the WSL CPU, 20 min on the A30).
• The regression case selfgravity-star freezes the equilibrium at ${16}^3$ in the ${10}^{-2}$ atmosphere without the limiter, not SV-7's light-atmosphere setup.
• Isolated self-gravity on AMR holds the star and keeps the exchange exact (SV-14). Refining the star halves its spurious motion against the uniform base, as a uniform grid of the fine spacing would; the collapse drifts at third order in $\mathrm{\Delta }t$ with the momentum at round-off, as on the uniform grid (SV-8): the James step is symmetric and the per-stage reflux carries the seam gravity work.
• Keep refinement boundaries off the stellar surface. With the SV-7 star ($a=1/2$) the box's faces lie on the surface; in the light atmosphere the end-of-step reflux run goes non-finite at $t=1.41$, the per-stage one reaches $t=2$ (#88). Production codes keep coarse–fine boundaries out of dense gas for the same reason (Castro refines everything far below the stellar density, Katz et al. 2016; Berger and Colella 1989 on the spurious waves of a resolution jump at a steep gradient).
• A runtime-AMR star changes its grid once, and needs conservative-state (SV-15). One level above the ${32}^3$ base can only refine the 8 inner blocks (0.5 wide), two levels carry the 2:1 balance to the boundary blocks (refused, D-87-4), and 4-cell blocks are below $2n_{gc}$: the collapse's first regrid refines the 8 inner blocks and later regrids keep that grid. With the default prolongation the collapse then drains a cell's internal energy to negative; conservative-state, one range limit for the whole state, runs it to the end (#88). In a sweep on WSL (CPU) it also ran the star of radius 0.25 regridded every 5 steps and of radius 0.3 every 8, on which the default goes non-finite (steps 53 and 48), and of radius 0.3 every 3, which the default survives; it fails on the radius-0.22 star regridded every 5 steps and the radius-0.25 star every 8 (3.5 and 4.0 cells per radius on the base), which the non-conservative linear prolongation runs to the end.
• Refine before the collapse: the Jeans criterion (SV-16). A runtime-AMR star prolonged mid-collapse is the fragile case; a Jeans marker refines it at the start instead, and the collapse never regrids. With 8 cells per Jeans length the radius-0.22 star regridded every 5 steps, which no conservative prolongation survives, reaches $t=1.5$ with the default one; on WSL so did the radius-0.25 star regridded every 8 and the SV-15 star. For this polytrope the Jeans length does not shrink in the collapse (${\lambda }_J=\sqrt{2}a$ at $\gamma =2$), so one refinement at the start holds; in an adiabatic collapse ${\lambda }_J\propto {\rho }^{(\gamma -2)/2}$, so for any $\gamma <2$ it shortens as the density rises, and the runtime regrids then follow it.
• Or carry the entropy: dual energy (SV-17). The coldest cells of the SV-15 collapse sit at $e/E\approx {10}^{-2}$ two steps before the default run goes non-finite; euler-de lifts them to their adiabat when they drop below it, and the run ends with the default prolongation. On WSL it also ran the radius-0.25 star regridded every 5 and every 8 steps; the radius-0.22 star (3.5 cells per radius) still needs the Jeans marker. Three remedies, one per cause: conservative-state keeps a mid-collapse regrid from building cold fast cells, the Jeans marker avoids the mid-collapse regrid, dual energy keeps the cold infall from draining.