MultiBlockMomentsFV

A classic finite-volume solver for the 2D 13-moment (Grad/Regularized-13) equations of rarefied gas dynamics on Cartesian multi-block meshes.

The scheme is a second-order MUSCL reconstruction with an HLL-type numerical flux, SSP-RK2 (Heun) time stepping, and a direct (relaxation) integration of the stiff production terms. The numerical method is described in: M. Torrilhon, "Two-Dimensional Bulk Microflow Simulations Based on Regularized Grad's 13-Moment Equations," Multiscale Modeling & Simulation, Vol. 5, No. 3, pp. 695–728, 2006. DOI: 10.1137/050635444.

Repository layout

MultiBlockMomentsFV/
├── Makefile          build / run / check / clean
├── src/              solver sources (.c/.h)
├── cases/            input decks ("block files")
│   ├── square4.txt        4-block square domain (standard test, t_end=0.3)
│   ├── squareSmall.txt    coarse version of square4 (quick smoke test)
│   └── Lshape.txt         3-block L-shaped domain (t_end=0.6)
├── reference/        golden results for `make check`
├── tools/            plot_solution.jl (plotting), compare_solution.py (regression check)
├── build/            compiled objects + binary  (git-ignored)
└── runs/             simulation output           (git-ignored)

Build

Requires a C compiler with the math library (gcc recommended).

make            # compiles src/*.c -> build/moments
make clean      # removes build/

Run

Each case writes into its own runs/<case>/ directory:

make run CASE=cases/squareSmall.txt          # quick test (~100 steps)
make run CASE=cases/square4.txt              # standard reference run
make run CASE=cases/Lshape.txt               # L-shaped domain

Pass extra solver flags via ARGS:

make run CASE=cases/square4.txt ARGS="-Kn 0.01 -CFL 0.9"

Or run the binary directly:

./build/moments -bfile cases/squareSmall.txt -out runs/squareSmall

Command-line flags

The physics/numerics parameters can also be set inside the block file (see below). A command-line flag overrides the value read from the file, so the file is the reproducible record while flags are handy for sweeps.

Flag	Meaning	Default
-bfile <file>	block/input file defining parameters + grid	blockfile.txt
-out <dir>	directory for output files (created if missing)	current dir
-Kn <value>	Knudsen number	0.001
-R13 <value>	regularization switch: <1 = Grad13, >=1 = Reg13	0.0
-CFL <value>	target CFL number for adaptive time stepping	0.95
-limiter <name>	slope limiter (see below)	harmonic
-sfile <file>	restart: load initial state from a previous output	(none)

Block file format

A block file has a self-describing key = value header of global parameters, followed by a line per block. Keys are optional and order-independent; missing keys keep their built-in defaults. Lines starting with # or // are comments, and a trailing # ... comment is allowed after a value.

# global parameters (all optional)
Kn      = 0.001
R13     = 0
CFL     = 0.95
limiter = harmonic      # none | average | minmod | superbee | harmonic
dt_out  = 0.05          # output cadence: equal intervals up to t_list ...
t_list   = 0.3           # ... up to this output time
#dt_out = 0             # 0 = no cadence: output exactly at the t_list time(s) below
#t_list  = 0.1 0.2 0.3   # with dt_out = 0, output at each of these times

Block data (Maximal NB blocks!):
number/xmin,xmax/ymin,ymax/nstx,nsty/leftbound,rightbound,bottombound,topbound/
1/-1.0,0.5/-1.0,0.2/15,12/0,2,-1,4/
...

• Output times: controlled by dt_out together with t_list (the t=0 initial condition is always written):
  • dt_out > 0 — equal intervals of dt_out up to the first t_list value; any further t_list entries are ignored. If t_list/dt_out is non-integer the last interval is shortened so the final output lands exactly on t_list.
  • dt_out = 0 — no cadence; output exactly at each time listed in t_list (so a single t_list gives outputs at t=0 and t=t_list, and t_list = 0.1 0.2 0.3 gives those three independent output times).
• limiter: selects the reconstruction slope limiter — none (first order), average (unlimited central), minmod, superbee, or harmonic (van-Leer-type, default).
• Per-block line: block number, x-range, y-range, cell counts nstx,nsty, and the four neighbor codes for left/right/bottom/top edges. This part keeps the original compact positional format.
• Boundary codes: 0 = extrapolation (open), -1 = solid wall (adiabatic), >0 = synchronize with that block number (internal multi-block interface).

The maximum number of blocks is NB in src/defs.h (currently 10); a block number outside [1, NB] is rejected with an error.

Output

Results are written as Tecplot ASCII .dat files named Grad13<Type>_nst<gridsize>_Kn<value>_tAll.dat, containing all output time levels as separate ZONEs. Variables per point:

x, y, rho, vx, vy, T, p, Pxx, Pxy, Pyy, qx, qy

View with Tecplot/ParaView, or plot in Julia:

julia tools/plot_solution.jl runs/squareSmall/Grad13Riemann2D_*_tAll.dat

Validation

make check                                   # default: square4 (full reference run)
make check CHECKCASE=cases/squareSmall.txt   # quick smoke check

make check runs the case and compares its Tecplot output against reference/<case>.ref.dat numerically via tools/compare_solution.py. The comparison ignores physically meaningless differences (signed zeros, whitespace, Tecplot STRANDID metadata) and fails (non-zero exit) on any value that exceeds the tolerance, so it works as a real regression gate.

Tolerances default to ABSTOL=1e-10, RELTOL=1e-8 and can be overridden:

make check ABSTOL=1e-8 RELTOL=1e-6

To refresh a golden file after an intentional change, regenerate it with make run CASE=cases/<case>.txt and copy the resulting *_tAll.dat to reference/<case>.ref.dat.

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(c) Manuel Torrilhon, ETH Zurich 2010 – RWTH Aachen 2026