Passive scalars

ID	Group	Milestone	Backends	Figure	Regression
PS (ps1 to ps4)	Euler	M10 (#92)	CPU, FNL	yes	sod-ps

Case and scope

[physics] passive_scalars = N carries up to four scalars $\rho s_k$ advected with the flow (composition, tracers, metallicity), on euler and on euler-de (numerics). A passive scalar must not act on the flow, must be conserved, and must follow the flow exactly where its specific value is uniform.

What the case catches:

• a scalar that acts on the flow: the flow fields of a run with scalars must be those of the same run without them, bitwise;
• a scalar flux inconsistent with the mass flux: a uniform specific value must stay uniform, through every regrid and reflux;
• a slot that leaks: the unused scalar slots must stay empty;
• a backend that disagrees, or a silent input.

Models and numerics

Item	Choice	Code
Model	euler or euler-de with passive_scalars = N: four more conserved variables (nv 9 or 10), the first N used	adam_flume_euler_ps_library, adam_flume_euler_de_ps_library (the body adam_flume_euler_library_body.INC)
Space, time	as Sod (V1) and runtime regrids: WENO5 characteristic flux splitting; each scalar a characteristic field of eigenvalue $u_n$	adam_flume_cpu_euler_ps_kernels.F90, FNL adam_flume_fnl_euler_ps_kernels.F90
Initial values	s1..sN per region (the first region whose box holds the cell), and per inflow face	set_initial_conditions, adam_flume_ic_object.F90

Initial and boundary conditions

Leg	Setup
ps1	the sod-x input with two scalars, $s=1$ left of the diaphragm and 0 right of it, on euler and on euler-de, each against the same run without scalars
ps2	passive_scalars = 5; passive_scalars = 1 with mhd-ideal
ps3	the ps1 runs of CPU and FNL
ps4	the vortex-regrid input (the isentropic vortex on a periodic quadtree, levels 1–3, a regrid every 5 steps, reflux on, 100 steps) with one scalar, $s_1=1$ everywhere

Input files

File	Role
src/tests/flume/verification/scalars/make_ps.py	writes the inputs (`--case sod
src/tests/flume/verification/scalars/check.sh	runs the legs (--legs ps1,ps2,ps3,ps4, default all but ps3; FLUME_EXE selects the backend; ps3 compares the work directories of both)
src/tests/flume/verification/scalars/ps_oracle.py	the bitwise comparison and the drift bound (ps1), the CPU–FNL rule (ps3), the uniform scalar (ps4)
src/tests/flume/regression/sod-ps/input.ini	the frozen ps1 euler case
make_doc_runs.sh <runs> scalars, make_doc_figures.py --only passive-scalars	the figure (2 runs, 2 minutes)

cd src/tests/flume/verification/scalars && ./check.sh                                  # CPU
FLUME_EXE=$PWD/../../../../../exe/adam_flume_fnl ./check.sh                           # FNL
./check.sh --legs ps3                                                                 # CPU against FNL

Results

Left: Sod at $t=0.2$, 50 cells. The scalar rides the contact (exact at $x=0.6855$), smeared over a few cells like the density's contact; WENO keeps it within $[-2.5\cdot {10}^{-4},1.0005]$, not strictly in $[0,1]$. Right: the regridded vortex with $s_1=1$ everywhere: the largest $|s_1-1|$ stays at ${10}^{-15}$ and the integral at round-off through the six regrids.

Leg	Asserted	Bound	CPU	FNL
ps1	the flow fields (and the entropy of euler-de) against the run without scalars	bitwise	bitwise equal	bitwise equal
ps1	each scalar integral	${10}^{-13}$	$1.1\cdot {10}^{-16}$	0
ps1	the unused slots 3 and 4	zero	zero	zero
ps2	passive_scalars = 5; with mhd-ideal	refused	refused	refused
ps3	per field, FNL against CPU, relative to the runs without scalars	$\le 2\times$ theirs, or ${10}^{-10}$	the flow fields as the base ($\le 6.3\cdot {10}^{-14}$), the scalars $2.2\cdot {10}^{-14}$	(the same comparison)
ps4	the largest $|s_1-1|$; the integral drift	${10}^{-12}$; ${10}^{-13}$	$1.6\cdot {10}^{-15}$; $2.2\cdot {10}^{-16}$	$1.0\cdot {10}^{-15}$; $1.1\cdot {10}^{-16}$

Related cases: Sod is the flow of ps1; runtime regrids the grid of ps4; the entropy of euler-de uses the same machinery (dual energy).

Findings

• A passive scalar is invisible to the flow. Each scalar is a characteristic field of its own, and the Euler waves' left eigenvectors have no component on it, so the flow fields of a run with scalars are bitwise those without.
• A uniform scalar stays uniform to round-off, through AMR. Its characteristic field $\rho s-s\rho$ vanishes, so the scalar flux is $s$ times the mass flux whatever the reconstruction; the prolongation, the restriction and the reflux treat it like the density.
• WENO bounds a scalar only to its own accuracy. A mass fraction across a contact over- and undershoots by $5\cdot {10}^{-4}$; a model that needs $s\in [0,1]$ strictly (chemistry) would need a bound-preserving limiter on the scalars.
• The fixed capacity costs width. The kernels carry four slots whatever $N$ (D-M10-1): a scalar run projects the wider state, the unused slots stay exactly zero.