---
title: Discrete invariants
summary: Summation by parts, free-stream preservation, conservation and entropy conservation are algebraic identities of the scheme. They hold to round-off, in two and three dimensions, on curved meshes.
section: verification
order: 2
level: 2
study: operators
studies: [free_stream, entropy_conservation, viscous_operator]
keywords: [SBP, free-stream preservation, conservation, entropy conservation, symmetry]
---
<!-- gyre docs: expanded Markdown of verification/discrete-invariants.md (generated tables, verification numbers and chart data are written out). Base level of this page: L2 Practice; passages at another level are bracketed by 'begin level N' / 'end level N' comments and start with [LN Name]. Built from revision ceb1f4b0be. -->

These studies do not measure convergence; they check identities that are either satisfied to
machine precision or broken.

## Summation-by-parts operators

**Verification status: Summation-by-parts operators** (study `operators`): **PASSED**, 1 of 1 checks passed, 0.29 s.

Gauss–Lobatto collocation operators: the SBP identity M D + (M D)^T = B, quadrature exactness to degree 2p-1, exact differentiation of degree-p polynomials, skew-symmetry of the flux-differencing matrix M S and consistency of the p-multigrid transfers.

| Check | Value | Expected | Result |
|---|---|---|---|
| largest defect over all identities and degrees | 6.2172e-15 | < 1e-11 | pass |

Provenance: gyre 0.4.0; revision `39979fcf90` (commit `39979fcf902ad6116eb550b881b4278d4599c8e2`); 2026-11-12T10:22:15-0500; device cpu; quick suite. **Revision check: Same code as this build.** Produced by 39979fcf902ad6116eb550b881b4278d4599c8e2. Every commit between it and the commit this site was built from (ceb1f4b0bed41de53e586df10858079f5a2bcdd8) changes only documentation text, evidence records or bookkeeping: the code is the same. Full record: [operators.json](../data/verification/operators.json).

**Table: Maximum defects per polynomial degree** (Summation-by-parts operators)

| p | SBP | quadrature (deg <= 2p-1) | differentiation | skew(M S) | Pi I - 1 |
|---|---|---|---|---|---|
| 1 | 0 | 0 | 0 | 0 | 2.2204e-16 |
| 2 | 0 | 2.2204e-16 | 0 | 0 | 4.9960e-16 |
| 3 | 3.3307e-16 | 5.5511e-17 | 8.8818e-16 | 4.4409e-16 | 0 |
| 4 | 1.1102e-15 | 3.3307e-16 | 1.7764e-15 | 8.8818e-16 | 4.4409e-16 |
| 5 | 1.3866e-15 | 2.2204e-16 | 2.6645e-15 | 1.9984e-15 | 4.4409e-16 |
| 6 | 2.2128e-15 | 1.9429e-16 | 4.4409e-15 | 1.5543e-15 | 4.4409e-15 |
| 7 | 1.8936e-15 | 1.1102e-16 | 3.1086e-15 | 2.2204e-15 | 1.8874e-15 |
| 8 | 2.5535e-15 | 4.4409e-16 | 6.2172e-15 | 3.7748e-15 | 1.1423e-15 |

Provenance: gyre 0.4.0; revision `39979fcf90` (commit `39979fcf902ad6116eb550b881b4278d4599c8e2`); 2026-11-12T10:22:15-0500; device cpu; quick suite. **Revision check: Same code as this build.** Produced by 39979fcf902ad6116eb550b881b4278d4599c8e2. Every commit between it and the commit this site was built from (ceb1f4b0bed41de53e586df10858079f5a2bcdd8) changes only documentation text, evidence records or bookkeeping: the code is the same. Full record: [operators.json](../data/verification/operators.json).

<!-- begin level 3 (Mechanics) -->
**[L3 Mechanics] What is being checked**

For every polynomial degree: the SBP identity $\mathbf M\mathbf D + (\mathbf M\mathbf D)^{\mathsf T} =
\mathbf B$; exactness of LGL quadrature up to degree $2p-1$ (and, in the unit tests, *inexactness*
at $2p$); exact differentiation of polynomials up to degree $p$; skew-symmetry of the
flux-differencing matrix $\mathbf M\mathbf S$; and that projection after prolongation is the identity
on the coarse space of the p-multigrid transfers.

<!-- end level 3 -->

## Free-stream preservation

**Verification status: Free-stream preservation on curved meshes** (study `free_stream`): **PASSED**, 1 of 1 checks passed, 18.67 s.

A uniform flow must remain an exact steady state on arbitrarily curved elements. This requires discrete metric identities: cross-product metrics in 2-D and the conservative-curl form in 3-D. Meshes are smoothly warped periodic boxes (6 % amplitude); the table reports the largest weak residual of the uniform state and the closure of each element's discrete surface (geometric conservation law).

| Check | Value | Expected | Result |
|---|---|---|---|
| max \|Rw\| of uniform flow and GCL defect | 2.9091e-15 | < 1e-11 | pass |

Provenance: gyre 0.4.0; revision `39979fcf90` (commit `39979fcf902ad6116eb550b881b4278d4599c8e2`); 2026-11-12T10:22:33-0500; device cpu; quick suite. **Revision check: Same code as this build.** Produced by 39979fcf902ad6116eb550b881b4278d4599c8e2. Every commit between it and the commit this site was built from (ceb1f4b0bed41de53e586df10858079f5a2bcdd8) changes only documentation text, evidence records or bookkeeping: the code is the same. Full record: [free_stream.json](../data/verification/free_stream.json).

**Table: Uniform-flow residual on warped meshes** (Free-stream preservation on curved meshes)

| dim | elements | p | geometry degree | viscous | min J / max J | max \|Rw\| | surface closure |
|---|---|---|---|---|---|---|---|
| 2-D | 6x5 | 3 | 4 | no | 0.451047 | 1.3492e-15 | 3.0531e-16 |
| 2-D | 6x5 | 3 | 4 | yes | 0.451047 | 1.3368e-15 | 3.0531e-16 |
| 2-D | 4x4 | 5 | 4 | no | 0.453556 | 2.8560e-15 | 4.9960e-16 |
| 2-D | 4x4 | 5 | 4 | yes | 0.453556 | 2.9091e-15 | 4.9960e-16 |
| 3-D | 3x3x3 | 2 | 2 | no | 0.276159 | 2.6316e-16 | 3.4694e-17 |
| 3-D | 3x3x3 | 2 | 2 | yes | 0.276159 | 2.6316e-16 | 3.4694e-17 |
| 3-D | 3x3x3 | 3 | 3 | no | 0.417089 | 3.2526e-16 | 6.9823e-17 |
| 3-D | 3x3x3 | 3 | 3 | yes | 0.417089 | 3.2823e-16 | 6.9823e-17 |

Provenance: gyre 0.4.0; revision `39979fcf90` (commit `39979fcf902ad6116eb550b881b4278d4599c8e2`); 2026-11-12T10:22:33-0500; device cpu; quick suite. **Revision check: Same code as this build.** Produced by 39979fcf902ad6116eb550b881b4278d4599c8e2. Every commit between it and the commit this site was built from (ceb1f4b0bed41de53e586df10858079f5a2bcdd8) changes only documentation text, evidence records or bookkeeping: the code is the same. Full record: [free_stream.json](../data/verification/free_stream.json).

<!-- begin level 3 (Mechanics) -->
**[L3 Mechanics] Why the 3-D rows matter**

In three dimensions the naive cross-product metrics break the discrete metric identities on
curved elements; the conservative curl form restores them. The 3-D rows of the table are
therefore a direct test of that construction. The last column is the closure of each element's
discrete surface, $\sum_\text{faces}\int \mathbf n\,\mathrm dS = 0$, the geometric conservation law.

<!-- end level 3 -->

## Conservation and entropy

**Verification status: Entropy conservation and stability** (study `entropy_conservation`): **PASSED**, 41 of 41 checks passed, 15.77 s.

The rate of change of the discrete mathematical entropy, -w^T Rw, is evaluated on warped periodic meshes for combinations of two-point volume fluxes and interface fluxes. The state is a periodic field carrying a 3 % node-wise perturbation, so it jumps across every element face, the curved interior faces included. Split-form DGSEM with an entropy-conservative two-point flux satisfies a discrete entropy balance exactly for any nodal state: the checks require the rate to vanish to round-off for the entropy-conservative pairings, to be clearly non-zero for the volume fluxes that are not entropy conservative, and to be negative, well above round-off, when an upwind interface flux adds dissipation. A second table tests the interface fluxes alone on random pairs of states far apart: the entropy-stable ones (llf, es_roe) must produce no entropy for any pair, which is the property the proof rests on; the classical solvers are shown for comparison.

| Check | Value | Expected | Result |
|---|---|---|---|
| 2-D ranocha / ec: \|d/dt entropy\| | 5.9781e-15 | < 1e-11 | pass |
| 2-D ranocha / ec: conservation defect | 4.4631e-14 | < 1e-11 | pass |
| 2-D chandrashekar / ec: \|d/dt entropy\| | 8.0969e-15 | < 1e-11 | pass |
| 2-D chandrashekar / ec: conservation defect | 4.8850e-14 | < 1e-11 | pass |
| 2-D kennedy_gruber / ec: not entropy conservative | 5.9810e-04 | > 1e-8 | pass |
| 2-D kennedy_gruber / ec: conservation defect | 4.2411e-14 | < 1e-11 | pass |
| 2-D central / ec: not entropy conservative | 4.5667e-04 | > 1e-8 | pass |
| 2-D central / ec: conservation defect | 4.3077e-14 | < 1e-11 | pass |
| 2-D ranocha / llf: entropy dissipated | -0.0550953 | < -1e-8 | pass |
| 2-D ranocha / llf: conservation defect | 4.6629e-14 | < 1e-11 | pass |
| 2-D ranocha / es_roe: entropy dissipated | -0.0183493 | < -1e-8 | pass |
| 2-D ranocha / es_roe: conservation defect | 4.4631e-14 | < 1e-11 | pass |
| 2-D chandrashekar / es_roe: entropy dissipated | -0.0183493 | < -1e-8 | pass |
| 2-D chandrashekar / es_roe: conservation defect | 4.7296e-14 | < 1e-11 | pass |
| 2-D ranocha / roe: entropy dissipated | -0.0183487 | < -1e-8 | pass |
| 2-D ranocha / roe: conservation defect | 4.1300e-14 | < 1e-11 | pass |
| 2-D ranocha / hllc: entropy dissipated | -0.0183448 | < -1e-8 | pass |
| 2-D ranocha / hllc: conservation defect | 4.0856e-14 | < 1e-11 | pass |
| 3-D ranocha / ec: \|d/dt entropy\| | 3.6880e-15 | < 1e-11 | pass |
| 3-D ranocha / ec: conservation defect | 1.1970e-14 | < 1e-11 | pass |
| 3-D chandrashekar / ec: \|d/dt entropy\| | 3.5232e-15 | < 1e-11 | pass |
| 3-D chandrashekar / ec: conservation defect | 1.1685e-14 | < 1e-11 | pass |
| 3-D kennedy_gruber / ec: not entropy conservative | 0.00119758 | > 1e-8 | pass |
| 3-D kennedy_gruber / ec: conservation defect | 2.6854e-15 | < 1e-11 | pass |
| 3-D central / ec: not entropy conservative | 0.00744448 | > 1e-8 | pass |
| 3-D central / ec: conservation defect | 1.8180e-15 | < 1e-11 | pass |
| 3-D ranocha / llf: entropy dissipated | -0.0543174 | < -1e-8 | pass |
| 3-D ranocha / llf: conservation defect | 1.1796e-14 | < 1e-11 | pass |
| 3-D ranocha / es_roe: entropy dissipated | -0.0177799 | < -1e-8 | pass |
| 3-D ranocha / es_roe: conservation defect | 1.1602e-14 | < 1e-11 | pass |
| 3-D chandrashekar / es_roe: entropy dissipated | -0.0177799 | < -1e-8 | pass |
| 3-D chandrashekar / es_roe: conservation defect | 1.2261e-14 | < 1e-11 | pass |
| 3-D ranocha / roe: entropy dissipated | -0.0178301 | < -1e-8 | pass |
| 3-D ranocha / roe: conservation defect | 1.3600e-15 | < 1e-11 | pass |
| 3-D ranocha / hllc: entropy dissipated | -0.0178184 | < -1e-8 | pass |
| 3-D ranocha / hllc: conservation defect | 8.1879e-16 | < 1e-11 | pass |
| 2-D llf: largest two-state entropy production over 5000 random pairs | -0.0571705 | < 0 | pass |
| 2-D es_roe: largest two-state entropy production over 5000 random pairs | -0.0301779 | < 0 | pass |
| 3-D llf: largest two-state entropy production over 5000 random pairs | -0.228196 | < 0 | pass |
| 3-D es_roe: largest two-state entropy production over 5000 random pairs | -0.0411714 | < 0 | pass |
| classical solvers (roe, hllc) violate the two-state inequality for some pairs (cases of 4) | 2 | >= 1 (why they carry no proof) | pass |

Provenance: gyre 0.4.0; revision `39979fcf90` (commit `39979fcf902ad6116eb550b881b4278d4599c8e2`); 2026-11-12T10:22:32-0500; device cpu; quick suite. **Revision check: Same code as this build.** Produced by 39979fcf902ad6116eb550b881b4278d4599c8e2. Every commit between it and the commit this site was built from (ceb1f4b0bed41de53e586df10858079f5a2bcdd8) changes only documentation text, evidence records or bookkeeping: the code is the same. Full record: [entropy_conservation.json](../data/verification/entropy_conservation.json).

**Chart: Entropy production rate by flux pairing** (Entropy conservation and stability). Bar chart of |d/dt ∫ η| against volume flux / interface flux, x axis category, y axis log; 2 series. The table holds the plotted data. Measured here: with an entropy-conservative volume flux and the dissipation-free interface flux the entropy rate is at most 8.1e-15 in magnitude; with the upwind interface fluxes it lies between -5.5e-02 and -1.8e-02.

| volume flux / interface flux | 2-D | 3-D |
|---|---|---|
| ranocha / ec | 5.9781e-15 | 3.6880e-15 |
| chandrashekar / ec | 8.0969e-15 | 3.5232e-15 |
| kennedy_gruber / ec | 5.9810e-04 | 0.00119758 |
| central / ec | 4.5667e-04 | 0.00744448 |
| ranocha / llf | 0.0550953 | 0.0543174 |
| ranocha / es_roe | 0.0183493 | 0.0177799 |
| chandrashekar / es_roe | 0.0183493 | 0.0177799 |
| ranocha / roe | 0.0183487 | 0.0178301 |
| ranocha / hllc | 0.0183448 | 0.0178184 |

Provenance: gyre 0.4.0; revision `39979fcf90` (commit `39979fcf902ad6116eb550b881b4278d4599c8e2`); 2026-11-12T10:22:32-0500; device cpu; quick suite. **Revision check: Same code as this build.** Produced by 39979fcf902ad6116eb550b881b4278d4599c8e2. Every commit between it and the commit this site was built from (ceb1f4b0bed41de53e586df10858079f5a2bcdd8) changes only documentation text, evidence records or bookkeeping: the code is the same. Full record: [entropy_conservation.json](../data/verification/entropy_conservation.json).

**Table: Entropy rate and conservation defect** (Entropy conservation and stability)

| dim | volume flux | interface flux | d/dt ∫ η | \|Σ Rw\| |
|---|---|---|---|---|
| 2-D | ranocha | ec | -5.9781e-15 | 4.4631e-14 |
| 2-D | chandrashekar | ec | -8.0969e-15 | 4.8850e-14 |
| 2-D | kennedy_gruber | ec | -5.9810e-04 | 4.2411e-14 |
| 2-D | central | ec | -4.5667e-04 | 4.3077e-14 |
| 2-D | ranocha | llf | -0.0550953 | 4.6629e-14 |
| 2-D | ranocha | es_roe | -0.0183493 | 4.4631e-14 |
| 2-D | chandrashekar | es_roe | -0.0183493 | 4.7296e-14 |
| 2-D | ranocha | roe | -0.0183487 | 4.1300e-14 |
| 2-D | ranocha | hllc | -0.0183448 | 4.0856e-14 |
| 3-D | ranocha | ec | -3.6880e-15 | 1.1970e-14 |
| 3-D | chandrashekar | ec | -3.5232e-15 | 1.1685e-14 |
| 3-D | kennedy_gruber | ec | -0.00119758 | 2.6854e-15 |
| 3-D | central | ec | -0.00744448 | 1.8180e-15 |
| 3-D | ranocha | llf | -0.0543174 | 1.1796e-14 |
| 3-D | ranocha | es_roe | -0.0177799 | 1.1602e-14 |
| 3-D | chandrashekar | es_roe | -0.0177799 | 1.2261e-14 |
| 3-D | ranocha | roe | -0.0178301 | 1.3600e-15 |
| 3-D | ranocha | hllc | -0.0178184 | 8.1879e-16 |

Provenance: gyre 0.4.0; revision `39979fcf90` (commit `39979fcf902ad6116eb550b881b4278d4599c8e2`); 2026-11-12T10:22:32-0500; device cpu; quick suite. **Revision check: Same code as this build.** Produced by 39979fcf902ad6116eb550b881b4278d4599c8e2. Every commit between it and the commit this site was built from (ceb1f4b0bed41de53e586df10858079f5a2bcdd8) changes only documentation text, evidence records or bookkeeping: the code is the same. Full record: [entropy_conservation.json](../data/verification/entropy_conservation.json).

<!-- begin level 2 (Practice) -->
**[L2 Practice] Reading the chart**

The state of the study is a smooth periodic field with a node-wise random perturbation of 3 %,
so every interface carries a jump and the interface flux is exercised on every face, curved
interior faces included.

With an entropy-conservative volume flux and a dissipation-free interface flux (`ec`), the
discrete entropy changes only at round-off level on a warped mesh — it is conserved. The
Kennedy–Gruber and central fluxes, which are not entropy conservative, produce or destroy
entropy at a rate many orders of magnitude larger (see the table). With upwind interface fluxes
(`llf`, `es_roe`, `roe`, `hllc`) the rate must be negative with a margin: the check requires a
rate below $-10^{-8}$, three orders above the tolerance of the conservative pairings, so that a
round-off value of either sign cannot pass. Entropy is dissipated, as the second law requires.
For small jumps the matrix dissipation of `es_roe` removes entropy at the rate of the
classical Roe flux, about a third of the scalar dissipation of `llf` in this study.

**Table: Entropy production of the interface flux alone, random state pairs (Ranocha two-point flux)** (Entropy conservation and stability)

| dim | interface flux | pairs | largest production | pairs with production > 0 [%] |
|---|---|---|---|---|
| 2-D | llf | 5000 | -0.0571705 | 0 |
| 2-D | es_roe | 5000 | -0.0301779 | 0 |
| 2-D | roe | 5000 | 1365.06 | 18.5 |
| 2-D | hllc | 5000 | -0.0156571 | 0 |
| 3-D | llf | 5000 | -0.228196 | 0 |
| 3-D | es_roe | 5000 | -0.0411714 | 0 |
| 3-D | roe | 5000 | 812.676 | 19.18 |
| 3-D | hllc | 5000 | -0.0401914 | 0 |

Provenance: gyre 0.4.0; revision `39979fcf90` (commit `39979fcf902ad6116eb550b881b4278d4599c8e2`); 2026-11-12T10:22:32-0500; device cpu; quick suite. **Revision check: Same code as this build.** Produced by 39979fcf902ad6116eb550b881b4278d4599c8e2. Every commit between it and the commit this site was built from (ceb1f4b0bed41de53e586df10858079f5a2bcdd8) changes only documentation text, evidence records or bookkeeping: the code is the same. Full record: [entropy_conservation.json](../data/verification/entropy_conservation.json).

The second table tests the interface fluxes alone, on random pairs of states far apart: the
entropy-stable fluxes produce no entropy for any pair, the classical Roe flux does for a
share of them ([why](../theory/two-point-fluxes.md#interface-fluxes)).
In every case mass, momentum and energy are conserved to round-off.

<!-- end level 2 -->

## Structure of the viscous operator

**Verification status: Structure of the BR2 viscous operator** (study `viscous_operator`): **PASSED**, 17 of 17 checks passed, 116.9 s.

At a quiescent state the energy–energy and momentum–momentum blocks of the weak-form Jacobian are exactly the discrete heat-conduction and viscous-stress operators. Their symmetry (the hallmark of an adjoint-consistent diffusion discretization) and positive semi-definiteness are checked on Cartesian and warped meshes with periodic and Dirichlet boundaries, and the Jacobian sparsity pattern is checked to be compact (face neighbours only), which element-block preconditioning relies on.

| Check | Value | Expected | Result |
|---|---|---|---|
| warp=0.0, periodic, p=2: asymmetry | 2.7046e-16 | < 1e-12 | pass |
| warp=0.0, periodic, p=2: smallest eigenvalue | -5.4470e-16 | >= -1e-12 | pass |
| warp=0.0, periodic, p=3: asymmetry | 2.5421e-16 | < 1e-12 | pass |
| warp=0.0, periodic, p=3: smallest eigenvalue | -1.8926e-15 | >= -1e-12 | pass |
| warp=0.0, Dirichlet, p=2: asymmetry | 1.1709e-16 | < 1e-12 | pass |
| warp=0.0, Dirichlet, p=2: smallest eigenvalue | 0.0252442 | >= -1e-12 | pass |
| warp=0.0, Dirichlet, p=3: asymmetry | 1.4417e-16 | < 1e-12 | pass |
| warp=0.0, Dirichlet, p=3: smallest eigenvalue | 0.0147011 | >= -1e-12 | pass |
| warp=0.05, periodic, p=2: asymmetry | 2.8210e-16 | < 1e-12 | pass |
| warp=0.05, periodic, p=2: smallest eigenvalue | 5.5369e-17 | >= -1e-12 | pass |
| warp=0.05, periodic, p=3: asymmetry | 2.9273e-16 | < 1e-12 | pass |
| warp=0.05, periodic, p=3: smallest eigenvalue | -5.0847e-16 | >= -1e-12 | pass |
| warp=0.05, Dirichlet, p=2: asymmetry | 1.5964e-16 | < 1e-12 | pass |
| warp=0.05, Dirichlet, p=2: smallest eigenvalue | 0.0247796 | >= -1e-12 | pass |
| warp=0.05, Dirichlet, p=3: asymmetry | 1.6369e-16 | < 1e-12 | pass |
| warp=0.05, Dirichlet, p=3: smallest eigenvalue | 0.0143994 | >= -1e-12 | pass |
| element couplings outside the face-neighbour stencil | 0 | == 0 | pass |

Provenance: gyre 0.4.0; revision `39979fcf90` (commit `39979fcf902ad6116eb550b881b4278d4599c8e2`); 2026-11-12T10:24:33-0500; device cpu; quick suite. **Revision check: Same code as this build.** Produced by 39979fcf902ad6116eb550b881b4278d4599c8e2. Every commit between it and the commit this site was built from (ceb1f4b0bed41de53e586df10858079f5a2bcdd8) changes only documentation text, evidence records or bookkeeping: the code is the same. Full record: [viscous_operator.json](../data/verification/viscous_operator.json).

**Table: Relative asymmetry and smallest eigenvalue** (Structure of the BR2 viscous operator)

| warp | boundaries | p | asym. (heat) | asym. (stress) | min eig (heat) | min eig (stress) |
|---|---|---|---|---|---|---|
| 0 | periodic | 2 | 2.7046e-16 | 2.0254e-16 | -5.4470e-16 | 6.5560e-16 |
| 0 | periodic | 3 | 2.2839e-16 | 2.5421e-16 | 1.1123e-16 | -1.8926e-15 |
| 0 | Dirichlet | 2 | 7.9058e-17 | 1.1709e-16 | 0.0423156 | 0.0252442 |
| 0 | Dirichlet | 3 | 6.5254e-17 | 1.4417e-16 | 0.0246366 | 0.0147011 |
| 0.05 | periodic | 2 | 1.9043e-16 | 2.8210e-16 | 1.5375e-15 | 5.5369e-17 |
| 0.05 | periodic | 3 | 2.3864e-16 | 2.9273e-16 | 1.1192e-15 | -5.0847e-16 |
| 0.05 | Dirichlet | 2 | 5.4250e-17 | 1.5964e-16 | 0.0417777 | 0.0247796 |
| 0.05 | Dirichlet | 3 | 1.3399e-16 | 1.6369e-16 | 0.0242548 | 0.0143994 |

Provenance: gyre 0.4.0; revision `39979fcf90` (commit `39979fcf902ad6116eb550b881b4278d4599c8e2`); 2026-11-12T10:24:33-0500; device cpu; quick suite. **Revision check: Same code as this build.** Produced by 39979fcf902ad6116eb550b881b4278d4599c8e2. Every commit between it and the commit this site was built from (ceb1f4b0bed41de53e586df10858079f5a2bcdd8) changes only documentation text, evidence records or bookkeeping: the code is the same. Full record: [viscous_operator.json](../data/verification/viscous_operator.json).

<!-- begin level 3 (Mechanics) -->
**[L3 Mechanics] Interpretation**

At a quiescent state the energy–energy block of the Jacobian is the discrete heat-conduction
operator and the momentum–momentum block is the discrete viscous-stress operator (including the
cross-derivative coupling). Both are symmetric to round-off and positive semi-definite, on
warped meshes and with Dirichlet walls. A separate check confirms that the full Jacobian only
couples face neighbours.

<!-- end level 3 -->