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Gyre/Verification

Discrete invariants

In one minuteSummation 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.

Written at L2 Practice goes to L3 Mechanics 1872 words Markdown

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

Summation-by-parts operators#

PassedSummation-by-parts operators1 of 1 checks passed · 0.29 s · quick suite

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.

largest defect over all identities and degrees6.22e-15expected < 1e-11pass

gyre 0.4.0 · 39979fcf90 · 2026-11-12 10:22 · cpu · quick suite · record (JSON)Same code as this build

Maximum defects per polynomial degreeSummation-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

gyre 0.4.0 · 39979fcf90 · 2026-11-12 10:22 · cpu · quick suite · record (JSON)Same code as this build

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.

Free-stream preservation#

PassedFree-stream preservation on curved meshes1 of 1 checks passed · 18.67 s · quick suite

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).

max |Rw| of uniform flow and GCL defect2.91e-15expected < 1e-11pass

gyre 0.4.0 · 39979fcf90 · 2026-11-12 10:22 · cpu · quick suite · record (JSON)Same code as this build

Uniform-flow residual on warped meshesFree-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

gyre 0.4.0 · 39979fcf90 · 2026-11-12 10:22 · cpu · quick suite · record (JSON)Same code as this build

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.

Conservation and entropy#

PassedEntropy conservation and stability41 of 41 checks passed · 15.77 s · quick suite

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.

2-D ranocha / ec: |d/dt entropy|5.98e-15expected < 1e-11pass
2-D ranocha / ec: conservation defect4.46e-14expected < 1e-11pass
2-D chandrashekar / ec: |d/dt entropy|8.10e-15expected < 1e-11pass
2-D chandrashekar / ec: conservation defect4.88e-14expected < 1e-11pass
2-D kennedy_gruber / ec: not entropy conservative5.98e-04expected > 1e-8pass
2-D kennedy_gruber / ec: conservation defect4.24e-14expected < 1e-11pass
2-D central / ec: not entropy conservative4.57e-04expected > 1e-8pass
2-D central / ec: conservation defect4.31e-14expected < 1e-11pass
2-D ranocha / llf: entropy dissipated-0.0551expected < -1e-8pass
2-D ranocha / llf: conservation defect4.66e-14expected < 1e-11pass
2-D ranocha / es_roe: entropy dissipated-0.01835expected < -1e-8pass
2-D ranocha / es_roe: conservation defect4.46e-14expected < 1e-11pass
2-D chandrashekar / es_roe: entropy dissipated-0.01835expected < -1e-8pass
2-D chandrashekar / es_roe: conservation defect4.73e-14expected < 1e-11pass
2-D ranocha / roe: entropy dissipated-0.01835expected < -1e-8pass
2-D ranocha / roe: conservation defect4.13e-14expected < 1e-11pass
2-D ranocha / hllc: entropy dissipated-0.01834expected < -1e-8pass
2-D ranocha / hllc: conservation defect4.09e-14expected < 1e-11pass
3-D ranocha / ec: |d/dt entropy|3.69e-15expected < 1e-11pass
3-D ranocha / ec: conservation defect1.20e-14expected < 1e-11pass
3-D chandrashekar / ec: |d/dt entropy|3.52e-15expected < 1e-11pass
3-D chandrashekar / ec: conservation defect1.17e-14expected < 1e-11pass
3-D kennedy_gruber / ec: not entropy conservative0.001198expected > 1e-8pass
3-D kennedy_gruber / ec: conservation defect2.69e-15expected < 1e-11pass
3-D central / ec: not entropy conservative0.007444expected > 1e-8pass
3-D central / ec: conservation defect1.82e-15expected < 1e-11pass
3-D ranocha / llf: entropy dissipated-0.05432expected < -1e-8pass
3-D ranocha / llf: conservation defect1.18e-14expected < 1e-11pass
3-D ranocha / es_roe: entropy dissipated-0.01778expected < -1e-8pass
3-D ranocha / es_roe: conservation defect1.16e-14expected < 1e-11pass
3-D chandrashekar / es_roe: entropy dissipated-0.01778expected < -1e-8pass
3-D chandrashekar / es_roe: conservation defect1.23e-14expected < 1e-11pass
3-D ranocha / roe: entropy dissipated-0.01783expected < -1e-8pass
3-D ranocha / roe: conservation defect1.36e-15expected < 1e-11pass
3-D ranocha / hllc: entropy dissipated-0.01782expected < -1e-8pass
3-D ranocha / hllc: conservation defect8.19e-16expected < 1e-11pass
2-D llf: largest two-state entropy production over 5000 random pairs-0.05717expected < 0pass
2-D es_roe: largest two-state entropy production over 5000 random pairs-0.03018expected < 0pass
3-D llf: largest two-state entropy production over 5000 random pairs-0.2282expected < 0pass
3-D es_roe: largest two-state entropy production over 5000 random pairs-0.04117expected < 0pass
classical solvers (roe, hllc) violate the two-state inequality for some pairs (cases of 4)2expected >= 1 (why they carry no proof)pass

gyre 0.4.0 · 39979fcf90 · 2026-11-12 10:22 · cpu · quick suite · record (JSON)Same code as this build

Entropy production rate by flux pairingEntropy conservation and stability

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.

Data table

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

gyre 0.4.0 · 39979fcf90 · 2026-11-12 10:22 · cpu · quick suite · record (JSON)Same code as this build

Entropy rate and conservation defectEntropy 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

gyre 0.4.0 · 39979fcf90 · 2026-11-12 10:22 · cpu · quick suite · record (JSON)Same code as this build

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.

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

gyre 0.4.0 · 39979fcf90 · 2026-11-12 10:22 · cpu · quick suite · record (JSON)Same code as this build

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). In every case mass, momentum and energy are conserved to round-off.

Structure of the viscous operator#

PassedStructure of the BR2 viscous operator17 of 17 checks passed · 116.9 s · quick suite

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.

warp=0.0, periodic, p=2: asymmetry2.70e-16expected < 1e-12pass
warp=0.0, periodic, p=2: smallest eigenvalue-5.45e-16expected >= -1e-12pass
warp=0.0, periodic, p=3: asymmetry2.54e-16expected < 1e-12pass
warp=0.0, periodic, p=3: smallest eigenvalue-1.89e-15expected >= -1e-12pass
warp=0.0, Dirichlet, p=2: asymmetry1.17e-16expected < 1e-12pass
warp=0.0, Dirichlet, p=2: smallest eigenvalue0.02524expected >= -1e-12pass
warp=0.0, Dirichlet, p=3: asymmetry1.44e-16expected < 1e-12pass
warp=0.0, Dirichlet, p=3: smallest eigenvalue0.0147expected >= -1e-12pass
warp=0.05, periodic, p=2: asymmetry2.82e-16expected < 1e-12pass
warp=0.05, periodic, p=2: smallest eigenvalue5.54e-17expected >= -1e-12pass
warp=0.05, periodic, p=3: asymmetry2.93e-16expected < 1e-12pass
warp=0.05, periodic, p=3: smallest eigenvalue-5.08e-16expected >= -1e-12pass
warp=0.05, Dirichlet, p=2: asymmetry1.60e-16expected < 1e-12pass
warp=0.05, Dirichlet, p=2: smallest eigenvalue0.02478expected >= -1e-12pass
warp=0.05, Dirichlet, p=3: asymmetry1.64e-16expected < 1e-12pass
warp=0.05, Dirichlet, p=3: smallest eigenvalue0.0144expected >= -1e-12pass
element couplings outside the face-neighbour stencil0expected == 0pass

gyre 0.4.0 · 39979fcf90 · 2026-11-12 10:24 · cpu · quick suite · record (JSON)Same code as this build

Relative asymmetry and smallest eigenvalueStructure 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

gyre 0.4.0 · 39979fcf90 · 2026-11-12 10:24 · cpu · quick suite · record (JSON)Same code as this build

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.