Streaming wave

Physical setup

The initial radiation field is a sinusoidal wave in a periodic cube of side length L. All photons stream in one fixed direction n, so the physical flux is F = cEn and the M1 pressure is P = Enn. There is no emission or flux damping in this test.

For an integer mode vector m, define n = m/|m|. The reviewed profile uses m = (2, −3, 1), a mean energy density of 1 and a sinusoidal amplitude of 0.5. Integer components make the phase periodic on each Cartesian face. These profile constants are defined in the solver source; the generic background/amplitude configuration entries do not set them.

The standard runner uses a uniform cubic grid with hydrodynamics and gravity disabled. Its CGS box spans −3 × 10¹⁰ to +3 × 10¹⁰ cm on each axis, with c = 2.99792458 × 10¹⁰ cm/s and default final time 4 s. Opt-in adaptive runs are identified below; N = INX × 2^level then describes their configured finest equivalent resolution. Each completed run below lists its recorded parameters.

Comparison and interpretation

This is an exact solution of the nonlinear M1 equations in their free-streaming limit. Constant-direction transport reduces to scalar advection:

At a cell center x, the reference is the exact volume average of the translated sinusoid, including its attenuation from averaging over a finite cell:

Here sinc(a) = sin(a)/a, with sinc(0) = 1. The same expression initializes the cells and supplies the comparison at the recorded final time. The phase uses the integer vector; only the flux direction is normalized.

What this test measures

Look for phase error, loss of amplitude, directional bias, and the resolution dependence of all four field norms. Smooth transport can expose the nominal accuracy of the scheme, although limiting near extrema and time integration affect the measured order. Periodic boundaries and zero sources should conserve the integrated energy and each integrated flux component. A small conservation residual does not by itself establish that the wave shape is accurate.

All resolutions

No completed run is available for this problem in this batch. Its description is available above; results will appear here as runs finish.

Errors, budgets, and movies

Every norm uses the full three-dimensional domain, with cell-volume weighting. Here e = numerical − reference and V is the domain volume:

Orders require two completed runs with matching physical time, configuration, units, executable, and origin. An exact zero error has no logarithmic order. The central slice and line profiles are visual diagnostics: the report slice is the cell layer at z = Δx/2, and its line profile is at y = z = Δx/2.

Conservation is assessed using the measured domain integrals and the signed cumulative boundary and source budgets:

B is outward transport and S is the source change. Conservation plots and this page's budget tables show fractions of the initial totals: Q(t)/D, [Q(0) − B(t) + S(t)]/D, and the signed residual R(t)/D. The denominator D = |Q(0)| is fixed for the entire history. It never grows with the current total or accumulated transport/source budgets. A negative initial flux therefore starts at −1, and the plotted fraction retains its sign.

A component whose initial net flux is zero or negligible cannot be divided by its own initial total. It instead uses the fixed scale D = c |E₀|, where E₀ is the initial integrated radiation energy. This fallback is labeled on its panels and in the table. “Negligible” means |Q(0)|/(c |E₀|) ≤ 64ε ≈ 1.421085 × 10⁻¹⁴, with ε the double-precision machine epsilon. This prevents roundoff left by cancellation in a symmetric field from becoming the denominator. For the initially cold Marshak case, the prescribed bath sets D = E_bath V for energy and D = c E_bath V for flux, with E_bath = 4/c. These scales remain fixed even though radiation enters the initially empty domain.

Raw histories and dimensional summary columns remain in physical units: energy integrals in erg, physical-flux integrals in erg cm/s for CGS runs. Dividing a flux integral by c² gives radiation momentum. Fraction-history CSV files are dimensionless except for their time column; summary files record each denominator and its basis. Maximum fractional residuals use the same fixed denominator as the final residual.

For runs labeled RSLA, these budgets check the modified radiation equations. They do not establish conservation of physical gas-plus-radiation energy. With reduced-speed factor α, the closed-system coupled invariants are Egas + Er/α and gas momentum + F/(αc²), with physical c in the closure. Compare multiple α values before interpreting an RSLA result as an approximation to physical-c evolution.

Movies show the numerical Silo states with a fixed color range for each movie. Recorded states are held for their actual simulation-time gaps; intermediate solutions are not interpolated. The 3D option shows the domain's exterior surface. Movies and plots retain their recorded physical units.

Where this problem comes from

The benchmark is the suite's analytic periodic-advection construction, implemented by streamingWave in octotiger/test_problems/radiation/profiles.hpp. Its formula follows directly from the free-streaming M1 closure. No specific published test was identified as the source of this exact mode and amplitude.

Radiation method context

Skinner & Ostriker (2013), “A Two-moment Radiation Hydrodynamics Module in Athena Using a Time-explicit Godunov Method,” ApJS 206, 21, provides context for the two-moment M1 system and source integration. For the radiation-only Gaussian and sphere cases, Euler uses backward-Euler damping: Fⁿ⁺¹ = F^*/(1 + cχΔt), where F^* is the flux after transport. This uses the algebra of their equations (42b)–(43) with θ = 1 and prescribed χ. Their paper neglects scattering; the suite reuses that damping formula for its prescribed scattering medium. ADER2 revision 6 uses the same two-stage Radau IIA local integration for all eight problems. Transport is evaluated on the old state, a predicted time average and a corrected time average, with reconstruction recomputed each time. Prescribed-medium constraints are applied within the implicit stages: the Gaussian and sphere tests hold the material fixed; Marshak holds density and velocity fixed while thermal energy evolves; streaming disables interaction. The coupled problems evolve gas and radiation together on one shared timestep. This implementation does not yet include gravity, and uniform stiff coupled second-order accuracy remains unresolved. Historical revision 5 used exponential affine-forcing solutions for prescribed scattering, SDIRK2 for stationary Marshak heating, and Radau stages for coupled hydro–radiation. Those saved results retain their original method revision and do not validate revision 6. Here the formulas use physical light speed; reduced-speed runs retain their recorded transport speed.

The solver's M1 implementation also cites Hanawa & Audit (2014), “Reformulation of the M1 model of radiative transfer,” JQSRT 145, 9–16. These references describe the numerical/physical framework; the exact test definitions are the source functions identified above.

Reproducing this batch

Descriptions document the reviewed source implementation. Older or edited solver versions can use different profile constants or averaging rules; these are not all encoded in run.ini. Use the run metadata, executable hash, and saved batch source hashes to identify the implementation behind a result. A source path or descriptive formula alone is not proof of the executable's contents.