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Documentation of the main branch

These pages follow Mallard's main branch, which is ahead of release 0.8.0: some features they describe come with the next release.

References

The numerical methods Mallard implements and the reference data it is validated against, grouped by topic. Each entry says where Mallard uses it. Techniques with no entry here (the transmissive image faces, the mirror ghost cells of TENO-E stencils, the deterministic face-flux accumulation, the stencil-driven halo depth) are Mallard's own and are described in Numerical methods. The design notes (distributed memory, chemistry) cite their own sources for work not yet implemented.

Reconstruction

  • B. van Leer, Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method, J. Comput. Phys. 32, 101–136 (1979). doi:10.1016/0021-9991(79)90145-1 Used in: MUSCL reconstruction (type = "MUSCL"; src/numerics/face_reconstruction.cpp).
  • T. J. Barth and D. C. Jespersen, The design and application of upwind schemes on unstructured meshes, AIAA Paper 89-0366, 27th Aerospace Sciences Meeting (1989). doi:10.2514/6.1989-366 Used in: the barth_jespersen MUSCL limiter.
  • V. Venkatakrishnan, Convergence to steady state solutions of the Euler equations on unstructured grids with limiters, J. Comput. Phys. 118, 120–130 (1995). doi:10.1006/jcph.1995.1084 Used in: the venkatakrishnan MUSCL limiter (the default) and its threshold (K h)³ (venkatakrishnan_K).
  • D. J. Mavriplis, Revisiting the least-squares procedure for gradient reconstruction on unstructured meshes, AIAA Paper 2003-3986, 16th AIAA Computational Fluid Dynamics Conference (2003). doi:10.2514/6.2003-3986 Used in: the inverse-distance-weighted least-squares gradients of MUSCL and of the viscous terms (src/numerics/gradient.h).
  • T. J. Barth and P. O. Frederickson, Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction, AIAA Paper 90-0013, 28th Aerospace Sciences Meeting (1990). doi:10.2514/6.1990-13 Used in: k-exact least-squares reconstruction (TENO-E polynomials; the quadratic fit of the viscous cell gradients).
  • L. Fu, X. Y. Hu and N. A. Adams, A family of high-order targeted ENO schemes for compressible-fluid simulations, J. Comput. Phys. 305, 333–359 (2016). doi:10.1016/j.jcp.2015.10.037 Used in: the TENO stencil selection with a sharp cutoff CT, on which TENO-E builds.
  • F. Ji, T. Liang and L. Fu, A class of new high-order finite-volume TENO schemes for hyperbolic conservation laws with unstructured meshes, J. Sci. Comput. 92, 61 (2022). doi:10.1007/s10915-022-01925-5 Used in: TENO on unstructured meshes; the fallback that renormalizes the small-stencil weights (TENO-E details).
  • T. Liang, W. Shyy and L. Fu, Efficient arbitrary-high-order TENO schemes with local adaptive dissipation for compressible flow simulation on unstructured meshes, J. Sci. Comput. 104, 1 (2025). doi:10.1007/s10915-025-02918-w Used in: TENO-E, the TENO reconstruction of orders 3 to 6: large central and sector stencils, the density-based troubled-cell indicator, characteristic-wise selection with the adaptive cutoff (src/numerics/teno.cpp, teno.h).
  • P. Tsoutsanis, V. A. Titarev and D. Drikakis, WENO schemes on arbitrary mixed-element unstructured meshes in three space dimensions, J. Comput. Phys. 230, 1585–1601 (2011). doi:10.1016/j.jcp.2010.11.023 Used in: building the TENO-E central stencil from neighbor layers sorted by centroid distance.
  • X. Zhang and C.-W. Shu, On maximum-principle-satisfying high order schemes for scalar conservation laws, J. Comput. Phys. 229, 3091–3120 (2010). doi:10.1016/j.jcp.2009.12.030 Used in: the scaling limiter of bound_preserving = true, which scales troubled-cell polynomials toward the cell average to keep density and pressure within the neighbors' range.

Riemann solvers

  • E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, 3rd ed., Springer (2009). doi:10.1007/b79761 Used in: the exact Riemann solver of the tests (test/exact_riemann.h), the HLLC star states (Toro's equations 10.37–10.39), the Euler eigenvectors of the characteristic decomposition and the Roe solver, and the spherical explosion test (§17.1.3).
  • V. V. Rusanov, The calculation of the interaction of non-stationary shock waves and obstacles, USSR Comput. Math. Math. Phys. 1, 304–320 (1962). doi:10.1016/0041-5553(62)90062-9 Used in: riemann_solver = "Rusanov".
  • A. Harten, P. D. Lax and B. van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev. 25, 35–61 (1983). doi:10.1137/1025002 Used in: riemann_solver = "HLL", and the HLL part of RHLL.
  • B. Einfeldt, On Godunov-type methods for gas dynamics, SIAM J. Numer. Anal. 25, 294–318 (1988). doi:10.1137/0725021
  • B. Einfeldt, C.-D. Munz, P. L. Roe and B. Sjögreen, On Godunov-type methods near low densities, J. Comput. Phys. 92, 273–295 (1991). doi:10.1016/0021-9991(91)90211-3 Used in: the wave-speed estimates (from Roe averages) of HLL and HLLC.
  • E. F. Toro, M. Spruce and W. Speares, Restoration of the contact surface in the HLL-Riemann solver, Shock Waves 4, 25–34 (1994). doi:10.1007/BF01414629 Used in: riemann_solver = "HLLC" (the default).
  • P. L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys. 43, 357–372 (1981). doi:10.1016/0021-9991(81)90128-5 Used in: riemann_solver = "Roe", the Roe part of RHLL, and the Roe averages of the wave-speed estimates.
  • A. Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys. 49, 357–393 (1983). doi:10.1016/0021-9991(83)90136-5 Used in: the entropy fix of the Roe solver on the acoustic waves.
  • P. Glaister, An approximate linearised Riemann solver for the Euler equations for real gases, J. Comput. Phys. 74, 382–408 (1988). doi:10.1016/0021-9991(88)90084-8 Used in: the Roe and RHLL solvers for gas mixtures, a Roe linearization for a general equation of state.
  • J.-S. Shuen, M.-S. Liou and B. van Leer, Inviscid flux-splitting algorithms for real gases with non-equilibrium chemistry, J. Comput. Phys. 90, 371–395 (1990). doi:10.1016/0021-9991(90)90172-W Used in: the Roe and RHLL solvers for gas mixtures: the Roe average of a multicomponent gas, whose composition waves move with the contact.
  • H. Nishikawa and K. Kitamura, Very simple, carbuncle-free, boundary-layer-resolving, rotated-hybrid Riemann solvers, J. Comput. Phys. 227, 2560–2581 (2008). doi:10.1016/j.jcp.2007.11.003 Used in: riemann_solver = "RHLL", the rotated hybrid of HLL and Roe.
  • K. H. Kim, C. Kim and O.-H. Rho, Methods for the accurate computations of hypersonic flows: I. AUSMPW+ scheme, J. Comput. Phys. 174, 38–80 (2001). doi:10.1006/jcph.2001.6873 Used in: RHLL's pressure weight min(p_l / p_r, p_r / p_l)^3, which blends the rotated flux toward HLL across faces that carry a shock.

  • J. J. Quirk, A contribution to the great Riemann solver debate, Int. J. Numer. Methods Fluids 18, 555–574 (1994). doi:10.1002/fld.1650180603 Used in: the carbuncle and odd–even decoupling tests that RHLL passes and HLLC and Roe fail (test/solver_test.cpp; the Shu–Osher validation).

  • H. Guillard and C. Viozat, On the behaviour of upwind schemes in the low Mach number limit, Comput. Fluids 28, 63–86 (1999). doi:10.1016/S0045-7930(98)00017-6 Used in: the motivation of the low-Mach correction: upwind dissipation that scales with the sound speed as M → 0.

  • F. Rieper, A low-Mach number fix for Roe's approximate Riemann solver, J. Comput. Phys. 230, 5263–5287 (2011). doi:10.1016/j.jcp.2011.03.025 Used in: the motivation of the low-Mach correction, and its normal-velocity-only variant that was compared against Thornber's.
  • B. Thornber, A. Mosedale, D. Drikakis, D. Youngs and R. J. R. Williams, An improved reconstruction method for compressible flows with low Mach number features, J. Comput. Phys. 227, 4873–4894 (2008). doi:10.1016/j.jcp.2008.01.036 Used in: the low-Mach correction of the face velocity jump, [numerics] low_mach_cutoff.
  • J. M. Weiss and W. A. Smith, Preconditioning applied to variable and constant density flows, AIAA J. 33, 2050–2057 (1995). doi:10.2514/3.12946 Used in: the cutoff Mach number of the low-Mach correction (low_mach_cutoff, default 0.1).

Viscous terms and transport

  • B. Diskin, J. L. Thomas, E. J. Nielsen, H. Nishikawa and J. A. White, Comparison of node-centered and cell-centered unstructured finite-volume discretizations: viscous fluxes, AIAA J. 48, 1326–1338 (2010). doi:10.2514/1.44940 Used in: the viscous face gradients, the average of the two cell gradients corrected along the face normal so that their component along the line of centroids equals the direct difference (src/numerics/viscous_flux.h).
  • W. Sutherland, The viscosity of gases and molecular force, Phil. Mag. (5) 36, 507–531 (1893). doi:10.1080/14786449308620508 Used in: viscosity_model = "sutherland".
  • C. R. Wilke, A viscosity equation for gas mixtures, J. Chem. Phys. 18, 517–519 (1950). doi:10.1063/1.1747673 Used in: the mixture viscosity of the mixture-averaged transport model, and its mixing operator in the thermal diffusion coefficients.
  • S. Mathur, P. K. Tondon and S. C. Saxena, Thermal conductivity of binary, ternary and quaternary mixtures of rare gases, Mol. Phys. 12, 569–579 (1967). doi:10.1080/00268976700100731 Used in: the mixture thermal conductivity, the mean of the mole-fraction-weighted arithmetic and harmonic means.

Large-eddy simulation

  • J. Smagorinsky, General circulation experiments with the primitive equations, Mon. Weather Rev. 91, 99–164 (1963). doi:10.1175/1520-0493(1963)091<0099:GCEWTP>2.3.CO;2 Used in: [les] model = "smagorinsky".
  • J. W. Deardorff, A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers, J. Fluid Mech. 41, 453–480 (1970). doi:10.1017/S0022112070000691 Used in: the LES filter width V^(1/3).
  • F. Nicoud and F. Ducros, Subgrid-scale stress modelling based on the square of the velocity gradient tensor, Flow Turbul. Combust. 62, 183–200 (1999). doi:10.1023/A:1009995426001 Used in: [les] model = "wale".
  • A. W. Vreman, An eddy-viscosity subgrid-scale model for turbulent shear flow: algebraic theory and applications, Phys. Fluids 16, 3670–3681 (2004). doi:10.1063/1.1785131 Used in: [les] model = "vreman".
  • F. Nicoud, H. Baya Toda, O. Cabrit, S. Bose and J. Lee, Using singular values to build a subgrid-scale model for large eddy simulations, Phys. Fluids 23, 085106 (2011). doi:10.1063/1.3623274 Used in: [les] model = "sigma" (the default in 3D), its constant and the closed-form singular values; the table of model properties checked in test/les_test.cpp.
  • M. Germano, U. Piomelli, P. Moin and W. H. Cabot, A dynamic subgrid-scale eddy viscosity model, Phys. Fluids A 3, 1760–1765 (1991). doi:10.1063/1.857955 Used in: [les] dynamic = true, the identity that sets the constant.
  • D. K. Lilly, A proposed modification of the Germano subgrid-scale closure method, Phys. Fluids A 4, 633–635 (1992). doi:10.1063/1.858280 Used in: [les] dynamic = true, the least-squares constant <L:M> / <M:M>.
  • P. Moin, K. Squires, W. Cabot and S. Lee, A dynamic subgrid-scale model for compressible turbulence and scalar transport, Phys. Fluids A 3, 2746–2757 (1991). doi:10.1063/1.858164 Used in: [les] dynamic = true, the Favre-weighted test-filtered stresses.
  • A. Scotti, C. Meneveau and D. K. Lilly, Generalized Smagorinsky model for anisotropic grids, Phys. Fluids A 5, 2306–2308 (1993). doi:10.1063/1.858537 Used in: [les] filter_width = "scotti".
  • B. Vreman, B. Geurts and H. Kuerten, Subgrid-modelling in LES of compressible flow, Appl. Sci. Res. 54, 191–203 (1995). doi:10.1007/BF00849116 Used in: the SGS terms of the filtered energy equation that are kept and neglected.
  • E. Garnier, N. Adams and P. Sagaut, Large Eddy Simulation for Compressible Flows, Springer (2009). doi:10.1007/978-90-481-2819-8 Used in: the Favre-filtered equations and the eddy-viscosity closure of the SGS heat flux.
  • S. Ghosal, An analysis of numerical errors in large-eddy simulations of turbulence, J. Comput. Phys. 125, 187–206 (1996). doi:10.1006/jcph.1996.0088 Used in: the motivation of the kinetic-energy budget ([integrals] budget), which separates the scheme's dissipation from the model's.
  • Y. Kuya, K. Totani and S. Kawai, Kinetic energy and entropy preserving schemes for compressible flows by split convective forms, J. Comput. Phys. 375, 823–853 (2018). doi:10.1016/j.jcp.2018.08.058 Used in: the central (KEEP) flux of convective_flux = "hybrid" (riemann::KEEP).
  • A. Jameson, Formulation of kinetic energy preserving conservative schemes for gas dynamics and direct numerical simulation of one-dimensional viscous compressible flow in a shock tube using entropy and kinetic energy preserving schemes, J. Sci. Comput. 34, 188–208 (2008). doi:10.1007/s10915-007-9172-6 Used in: the condition on the momentum flux under which the central flux changes the kinetic energy only by the pressure work (tested in test/les_test.cpp).
  • F. Ducros, V. Ferrand, F. Nicoud, C. Weber, D. Darracq, C. Gacherieu and T. Poinsot, Large-eddy simulation of the shock/turbulence interaction, J. Comput. Phys. 152, 517–549 (1999). doi:10.1006/jcph.1999.6238 Used in: the shock sensor of the hybrid flux.
  • A. Bhagatwala and S. K. Lele, A modified artificial viscosity approach for compressible turbulence simulations, J. Comput. Phys. 228, 4965–4969 (2009). doi:10.1016/j.jcp.2009.04.009 Used in: restricting the Ducros sensor to compressions, so expansions and flames keep the central flux.
  • O. Colin, F. Ducros, D. Veynante and T. Poinsot, A thickened flame model for large eddy simulations of turbulent premixed combustion, Phys. Fluids 12, 1843–1863 (2000). doi:10.1063/1.870436 Used in: [les.combustion] model = "tfles", and its subgrid velocity 2 Delta^3 |lap(curl u)|.
  • F. Charlette, C. Meneveau and D. Veynante, A power-law flame wrinkling model for LES of premixed turbulent combustion. Part I: non-dynamic formulation and initial tests, Combust. Flame 131, 159–180 (2002). doi:10.1016/S0010-2180(02)00400-5 Used in: the efficiency E of the thickened flame (efficiency = "charlette"), and its saturated power law E = F^beta (efficiency = "dynamic", the default).
  • F. Charlette, C. Meneveau and D. Veynante, A power-law flame wrinkling model for LES of premixed turbulent combustion. Part II: dynamic formulation, Combust. Flame 131, 181–197 (2002). doi:10.1016/S0010-2180(02)00401-7 Used in: the Germano-like identity for the flame surface that sets beta (efficiency = "dynamic").
  • G. Wang, M. Boileau and D. Veynante, Implementation of a dynamic thickened flame model for large eddy simulations of turbulent premixed combustion, Combust. Flame 158, 2199–2213 (2011). doi:10.1016/j.combustflame.2011.04.008 Used in: the dynamic efficiency of the thickened flame, with the combustion filter width F delta_L and a global beta (efficiency = "dynamic"; src/solver/solver_les.cpp).
  • P. S. Volpiani, T. Schmitt and D. Veynante, A posteriori tests of a dynamic thickened flame model for large eddy simulations of turbulent premixed combustion, Combust. Flame 174, 166–178 (2016). doi:10.1016/j.combustflame.2016.08.007 Used in: the dynamic efficiency as the default, and its a posteriori test against filtered DNS over several filter widths (design/les.md, section 8.3).
  • J.-P. Légier, T. Poinsot, B. Varoquié, F. Lacas and D. Veynante, Large eddy simulation of a non-premixed turbulent burner using a dynamically thickened flame model, in Advances in LES of Complex Flows, Springer, 315–326 (2002). doi:10.1007/978-94-017-1998-8_27 Used in: thickening only where the flame sensor is on (dynamic thickening).
  • T. D. Butler and P. J. O'Rourke, A numerical method for two dimensional unsteady reacting flows, Proc. Combust. Inst. 16, 1503–1515 (1977). doi:10.1016/S0082-0784(77)80432-3 Used in: the thickening transformation D -> F D, omega -> omega / F that keeps the laminar flame speed.

Boundary conditions

  • V. Sabelnikov and C. Fureby, LES combustion modeling for high Re flames using a multi-phase analogy, Combust. Flame 160, 83–96 (2013). doi:10.1016/j.combustflame.2012.09.008 Used in: the partially stirred reactor ([les.combustion] model = "pasr"), its reacting fraction kappa = tau_c / (tau_c + tau_mix).
  • T. J. Poinsot and S. K. Lele, Boundary conditions for direct simulations of compressible viscous flows, J. Comput. Phys. 101, 104–129 (1992). doi:10.1016/0021-9991(92)90046-2 Used in: the LODI relations of the characteristic boundaries (nscbc_outlet, nscbc_inlet; BoundaryData::characteristic_W in src/boundary/boundary.h) and their viscous outflow conditions.
  • D. H. Rudy and J. C. Strikwerda, A nonreflecting outflow boundary condition for subsonic Navier-Stokes calculations, J. Comput. Phys. 36, 55–70 (1980). doi:10.1016/0021-9991(80)90174-6 Used in: the pressure relaxation of nscbc_outlet and its default sigma.
  • L. Selle, F. Nicoud and T. Poinsot, Actual impedance of nonreflecting boundary conditions: implications for computation of resonators, AIAA J. 42(5), 958–964 (2004). doi:10.2514/1.1883 Used in: the reflection coefficient of the relaxed outlet, against which the acoustic-pulse test checks nscbc_outlet.
  • C. S. Yoo, Y. Wang, A. Trouvé and H. G. Im, Characteristic boundary conditions for direct simulations of turbulent counterflow flames, Combust. Theory Model. 9(4), 617–646 (2005). doi:10.1080/13647830500307378 Used in: the transverse terms of the characteristic boundaries.
  • C. S. Yoo and H. G. Im, Characteristic boundary conditions for simulations of compressible reacting flows with multi-dimensional, viscous and reaction effects, Combust. Theory Model. 11(2), 259–286 (2007). doi:10.1080/13647830600898995 Used in: the transverse terms and the treatment of multicomponent, reacting flows at characteristic boundaries.
  • G. Lodato, P. Domingo and L. Vervisch, Three-dimensional boundary conditions for direct and large-eddy simulation of compressible viscous flows, J. Comput. Phys. 227, 5105–5143 (2008). doi:10.1016/j.jcp.2008.01.038 Used in: the relaxation of the transverse terms with beta equal to the Mach number (beta).
  • V. Granet, O. Vermorel, T. Leonard, L. Gicquel and T. Poinsot, Comparison of nonreflecting outlet boundary conditions for compressible solvers on unstructured grids, AIAA J. 48(10), 2348–2364 (2010). doi:10.2514/1.J050391 Used in: the default beta and the vortex-outflow validation of nscbc_outlet.
  • M. Esmaily Moghadam, Y. Bazilevs, T.-Y. Hsia, I. E. Vignon-Clementel and A. L. Marsden, A comparison of outlet boundary treatments for prevention of backflow divergence with relevance to blood flow simulations, Comput. Mech. 48, 277–291 (2011). doi:10.1007/s00466-011-0599-0 Used in: the backflow stabilization tried, and not adopted, for nscbc_outlet (docs/design/nscbc.md).
  • D. J. Bodony, Analysis of sponge zones for computational fluid mechanics, J. Comput. Phys. 212, 681–702 (2006). doi:10.1016/j.jcp.2005.07.014 Used in: the sponge layers ([[sponges]]; src/solver/solver_sponge.cpp).
  • A. Mani, Analysis and optimization of numerical sponge layers as a nonreflective boundary treatment, J. Comput. Phys. 231, 704–716 (2012). doi:10.1016/j.jcp.2011.10.017 Used in: the sponge layers and the advice to ramp their strength.

Inflow turbulence

  • M. Klein, A. Sadiki and J. Janicka, A digital filter based generation of inflow data for spatially developing direct numerical or large eddy simulations, J. Comput. Phys. 186(2), 652–665 (2003). doi:10.1016/S0021-9991(03)00090-1 Used in: the digital filter of [boundaries.turbulence] (Gaussian kernels and their widths; src/boundary/synthetic_inflow.cpp).
  • T. S. Lund, X. Wu and K. D. Squires, Generation of turbulent inflow data for spatially-developing boundary layer simulations, J. Comput. Phys. 140(2), 233–258 (1998). doi:10.1006/jcph.1998.5882 Used in: the Cholesky factor that gives the synthetic fluctuations their Reynolds stress; recycling-rescaling, weighed in synthetic_inflow.md.
  • E. Touber and N. D. Sandham, Large-eddy simulation of low-frequency unsteadiness in a turbulent shock-induced separation bubble, Theor. Comput. Fluid Dyn. 23(2), 79–107 (2009). doi:10.1007/s00162-009-0103-z Used in: the digital filter in compressible simulations (design comparison).
  • Z.-T. Xie and I. P. Castro, Efficient generation of inflow conditions for large eddy simulation of street-scale flows, Flow Turbul. Combust. 81(3), 449–470 (2008). doi:10.1007/s10494-008-9151-5 Used in: design comparison (the forward-stepwise, exponential time correlation that Mallard does not use).
  • A. Kempf, M. Klein and J. Janicka, Efficient generation of initial- and inflow-conditions for transient turbulent flows in arbitrary geometries, Flow Turbul. Combust. 74(1), 67–84 (2005). doi:10.1007/s10494-005-3140-8 Used in: design comparison (digital filtering in arbitrary geometries).
  • N. Jarrin, S. Benhamadouche, D. Laurence and R. Prosser, A synthetic-eddy-method for generating inflow conditions for large-eddy simulations, Int. J. Heat Fluid Flow 27(4), 585–593 (2006). doi:10.1016/j.ijheatfluidflow.2006.02.006 Used in: design comparison (synthetic eddy method).
  • R. Poletto, T. Craft and A. Revell, A new divergence free synthetic eddy method for the reproduction of inlet flow conditions for LES, Flow Turbul. Combust. 91(3), 519–539 (2013). doi:10.1007/s10494-013-9488-2 Used in: design comparison (divergence-free synthetic eddies).
  • Y. Kim, I. P. Castro and Z.-T. Xie, Divergence-free turbulence inflow conditions for large-eddy simulations with incompressible flow solvers, Comput. Fluids 84, 56–68 (2013). doi:10.1016/j.compfluid.2013.06.001 Used in: design comparison (the dilatation of digital-filter fields and its pressure fluctuations).
  • N. Guézennec and T. Poinsot, Acoustically nonreflecting and reflecting boundary conditions for vorticity injection in compressible solvers, AIAA J. 47(7), 1709–1722 (2009). doi:10.2514/1.41749 Used in: the incoming wave of an inlet whose target varies in time, -rho c du_n,t/dt (CharacteristicStateFunctor in src/numerics/characteristic.h).
  • A. Keating, U. Piomelli, E. Balaras and H.-J. Kaltenbach, A priori and a posteriori tests of inflow conditions for large-eddy simulation, Phys. Fluids 16(12), 4696–4712 (2004). doi:10.1063/1.1811672 Used in: the development length of synthetic inflow in a channel, against which the channel validation compares.
  • X. Wu, Inflow turbulence generation methods, Annu. Rev. Fluid Mech. 49, 23–49 (2017). doi:10.1146/annurev-fluid-010816-060322 Used in: the comparison of inflow methods in synthetic_inflow.md.
  • G. Comte-Bellot and S. Corrsin, The use of a contraction to improve the isotropy of grid-generated turbulence, J. Fluid Mech. 25(4), 657–682 (1966). doi:10.1017/S0022112066000338 Used in: the decay exponent of grid turbulence, against which the duct validation compares.
  • G. K. Batchelor and A. A. Townsend, Decay of turbulence in the final period, Proc. R. Soc. Lond. A 194(1039), 527–543 (1948). doi:10.1098/rspa.1948.0095 Used in: the final-period decay exponent, the low-Reynolds-number bound of the duct validation.

Time integration and time step

  • C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys. 77, 439–471 (1988). doi:10.1016/0021-9991(88)90177-5
  • S. Gottlieb and C.-W. Shu, Total variation diminishing Runge–Kutta schemes, Math. Comp. 67, 73–85 (1998). doi:10.1090/S0025-5718-98-00913-2 Used in: time_integrator = "SSPRK3" (the default; src/numerics/time_integrator.cpp).
  • W. Kutta, Beitrag zur näherungsweisen Integration totaler Differentialgleichungen, Z. Math. Phys. 46, 435–453 (1901). Used in: time_integrator = "RK4", the classical fourth-order Runge–Kutta method.
  • J. Blazek, Computational Fluid Dynamics: Principles and Applications, 3rd ed., Butterworth-Heinemann (2015). doi:10.1016/C2013-0-19038-1 Used in: the local time step from convective and viscous spectral radii (Solver::calc_dt_cfl1), and the characteristic far-field condition (type = "farfield"), which takes the outgoing Riemann invariant from the interior and the incoming one from the free stream.

Quadrature and geometry

  • D. A. Dunavant, High degree efficient symmetrical Gaussian quadrature rules for the triangle, Int. J. Numer. Methods Eng. 21, 1129–1148 (1985). doi:10.1002/nme.1620210612 Used in: cell averages of analytical initial conditions, TENO-E integrals in 2D, and face quadrature on triangular faces in 3D (src/numerics/quadrature.cpp).
  • M. G. Duffy, Quadrature over a pyramid or cube of integrands with a singularity at a vertex, SIAM J. Numer. Anal. 19, 1260–1262 (1982). doi:10.1137/0719090
  • A. H. Stroud, Approximate Calculation of Multiple Integrals, Prentice-Hall (1971). Used in: collapsed (conical product) Gauss rules on triangles and tetrahedra, for TENO-E moments and 3D analytical initial conditions (src/numerics/teno.cpp, src/solver/solver_initialize.cpp). Gauss–Legendre rules integrate the faces.
  • H. W. Kuhn, Some combinatorial lemmas in topology, IBM J. Res. Dev. 4, 518–524 (1960). doi:10.1147/rd.45.0518 Used in: type = "cartesian_tet", six tetrahedra per block of a generated box.

Parallelism, meshes and software

  • J. Skilling, Programming the Hilbert curve, AIP Conf. Proc. 707, 381–387 (2004). doi:10.1063/1.1751381 Used in: partitioner = "hilbert" (src/parallel/partition.cpp).
  • M. Deveci, S. Rajamanickam, K. D. Devine and Ü. V. Çatalyürek, Multi-jagged: A scalable parallel spatial partitioning algorithm, IEEE Trans. Parallel Distrib. Syst. 27, 803–817 (2016). doi:10.1109/TPDS.2015.2412545 Used in: partitioner = "multijagged" (src/parallel/partition.cpp).
  • P. Sanders and D. Seemaier, Distributed deep multilevel graph partitioning, in Euro-Par 2023: Parallel Processing, Lecture Notes in Computer Science, Springer, 443–457 (2023). doi:10.1007/978-3-031-39698-4_30 Used in: partitioner = "graph" through dKaMinPar (KaMinPar; Mallard_ENABLE_KAMINPAR).
  • C. R. Trott et al., Kokkos 3: Programming model extensions for the exascale era, IEEE Trans. Parallel Distrib. Syst. 33, 805–817 (2022). doi:10.1109/TPDS.2021.3097283 Used in: all of Mallard's kernels and data (Kokkos).
  • C. Geuzaine and J.-F. Remacle, Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities, Int. J. Numer. Methods Eng. 79, 1309–1331 (2009). doi:10.1002/nme.2579 Used in: the Gmsh 2.2 and 4.1 mesh reader, and the meshes of tools/make_cylinder_mesh.py, tools/make_mallard_mesh.py and tools/make_sphere_re300_mesh.py.
  • toml11 reads the input file; exprtk evaluates its expressions.

Chemistry and multicomponent flow

  • R. J. Kee, M. E. Coltrin and P. Glarborg, Chemically Reacting Flow: Theory and Practice, Wiley (2003). doi:10.1002/0471461296 Used in: the thermally perfect mixture (NASA polynomials), rates of progress and equilibrium constants, the mixture-averaged transport model with its correction velocity ([chemistry], src/chemistry).
  • D. G. Goodwin, R. L. Speth, H. K. Moffat and B. W. Weber, Cantera: An object-oriented software toolkit for chemical kinetics, thermodynamics, and transport processes, https://www.cantera.org. doi:10.5281/zenodo.742000 Used in: the mechanism file format (YAML) Mallard reads, the transport-property fits ported from Cantera, and the reference data of the chemistry tests and validation (generated by scripts in tools/; Mallard does not link Cantera).
  • S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, 3rd ed., Cambridge University Press (1970).
  • T. Zirwes and A. Kronenburg, Assessment of approximate Soret diffusion models for hydrogen and ammonia combustion, Flow Turbul. Combust. 115, 1631–1650 (2025). doi:10.1007/s10494-025-00680-5
  • L. Monchick and E. A. Mason, Transport properties of polar gases, J. Chem. Phys. 35, 1676–1697 (1961). doi:10.1063/1.1732130 Used in (these three): the mixture-averaged thermal diffusion coefficients of physics.soret (Cantera's model, with its fits of the collision integral C*).
  • G. Dixon-Lewis, Flame structure and flame reaction kinetics II. Transport phenomena in multicomponent systems, Proc. R. Soc. Lond. A 307, 111–135 (1968). doi:10.1098/rspa.1968.0178
  • A. Ern and V. Giovangigli, Multicomponent Transport Algorithms, Lecture Notes in Physics Monographs m24, Springer (1994). doi:10.1007/978-3-540-48650-3
  • A. Ern and V. Giovangigli, Thermal diffusion effects in hydrogen-air and methane-air flames, Combust. Theory Model. 2, 349–372 (1998). doi:10.1088/1364-7830/2/4/001 Used in (these three): the assessment of full multicomponent transport (Cantera's L-matrix model) in docs/design/chemistry.md.
  • R. G. Gilbert, K. Luther and J. Troe, Theory of thermal unimolecular reactions in the fall-off range. II. Weak collision rate constants, Ber. Bunsenges. Phys. Chem. 87, 169–177 (1983). doi:10.1002/bbpc.19830870218 Used in: Troe falloff reactions.
  • P. H. Stewart, C. W. Larson and D. M. Golden, Pressure and temperature dependence of reactions proceeding via a bound complex. 2. Application to 2CH3 → C2H5 + H, Combust. Flame 75, 25–31 (1989). doi:10.1016/0010-2180(89)90084-9 Used in: SRI falloff reactions.
  • E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, 2nd ed., Springer (1996). doi:10.1007/978-3-642-05221-7 Used in: the RODAS Rosenbrock integrator of the chemistry source and MallardReactor.
  • G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal. 5, 506–517 (1968). doi:10.1137/0705041 Used in: the Strang splitting of chemistry around the flow step.
  • H. Wu, P. C. Ma and M. Ihme, Efficient time-stepping techniques for simulating turbulent reactive flows with stiff chemistry, Comput. Phys. Commun. 243, 81–96 (2019). doi:10.1016/j.cpc.2019.04.016 Used in: SIMPLER balanced splitting of chemistry and transport ([chemistry] coupling = "simpler").
  • B. Larrouturou, How to preserve the mass fractions positivity when computing compressible multi-component flows, J. Comput. Phys. 95, 59–84 (1991). doi:10.1016/0021-9991(91)90253-H Used in: the species fluxes (mass-flux upwinding of the reconstructed mass fractions).
  • R. Abgrall and S. Karni, Computations of compressible multifluids, J. Comput. Phys. 169, 594–623 (2001). doi:10.1006/jcph.2000.6685
  • G. Billet and R. Abgrall, An adaptive shock-capturing algorithm for solving unsteady reactive flows, Comput. Fluids 32, 1473–1495 (2003). doi:10.1016/S0045-7930(03)00004-5
  • P. C. Ma, Y. Lv and M. Ihme, An entropy-stable hybrid scheme for simulations of transcritical real-fluid flows, J. Comput. Phys. 340, 330–357 (2017). doi:10.1016/j.jcp.2017.03.022 Used in (these three): the double-flux option for oscillation-free material and temperature interfaces.
  • R. S. Barlow, A. N. Karpetis, J. H. Frank and J.-Y. Chen, Scalar profiles and NO formation in laminar opposed-flow partially premixed methane/air flames, Combust. Flame 127, 2102–2118 (2001). doi:10.1016/S0010-2180(01)00313-3
  • W. L. Grosshandler, RADCAL: a narrow-band model for radiation calculations in a combustion environment, NIST Technical Note 1402 (1993). doi:10.6028/NIST.TN.1402 Used in (these two): the optically thin radiation model of the TNF workshop, [radiation] (the Planck-mean absorption coefficients of H2O, CO2, CO and CH4 fitted to RADCAL; src/chemistry/radiation.h).
  • B. Franzelli, E. Riber, L. Y. M. Gicquel and T. Poinsot, Large Eddy Simulation of combustion instabilities in a lean partially premixed swirled flame, Combust. Flame 159, 621–637 (2012). doi:10.1016/j.combustflame.2011.08.004 Used in: mechanisms/ch4_bfer.yaml, the two-step mechanism 2S-CH4-BFER (lean branch) of examples/flame_vortex_quenching.

Validation cases and reference data

  • G. Comte-Bellot and S. Corrsin, Simple Eulerian time correlation of full- and narrow-band velocity signals in grid-generated, 'isotropic' turbulence, J. Fluid Mech. 48, 273–337 (1971). doi:10.1017/S0022112071001599

  • T. Saad, D. Cline, R. Stoll and J. C. Sutherland, Scalable tools for generating synthetic isotropic turbulence with arbitrary spectra, AIAA J. 55, 327–331 (2017). doi:10.2514/1.J055230 Used in (these two): examples/cbc_les (LES of decaying grid turbulence); the measured spectra at the three stations as tabulated with the second's TurboGenPY (examples/cbc_les/reference/cbc_spectra.csv, in SI units).

  • R. P. Fedkiw, B. Merriman and S. Osher, High accuracy numerical methods for thermally perfect gas flows with chemistry, J. Comput. Phys. 132, 175–190 (1997). doi:10.1006/jcph.1996.5622

  • P. J. Martínez Ferrer, R. Buttay, G. Lehnasch and A. Mura, A detailed verification procedure for compressible reactive multicomponent Navier–Stokes solvers, Comput. Fluids 89, 88–110 (2014). doi:10.1016/j.compfluid.2013.10.014 Used in (these two): examples/reactive_shock_tube (the reflected-shock ignition of 2H2–O2–7Ar).
  • Shock and Detonation Toolbox (SDToolbox), Explosion Dynamics Laboratory, California Institute of Technology, https://shepherd.caltech.edu/EDL/PublicResources/sdt/ Used in: the CJ speed and ZND structure of examples/detonation_1d (tools/detonation_reference.py follows its method with Cantera).
  • G. A. Sod, A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys. 27, 1–31 (1978). doi:10.1016/0021-9991(78)90023-2 Used in: examples/sod and the shock-tube tests.
  • C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, II, J. Comput. Phys. 83, 32–78 (1989). doi:10.1016/0021-9991(89)90222-2 Used in: examples/shu_osher.
  • C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics 1697, Springer, 325–432 (1998). doi:10.1007/BFb0096355 Used in: the isentropic vortex of the design-order convergence study.
  • P. Woodward and P. Colella, The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys. 54, 115–173 (1984). doi:10.1016/0021-9991(84)90142-6 Used in: examples/double_mach.
  • C. W. Schulz-Rinne, J. P. Collins and H. M. Glaz, Numerical solution of the Riemann problem for two-dimensional gas dynamics, SIAM J. Sci. Comput. 14, 1394–1414 (1993). doi:10.1137/0914082
  • P. D. Lax and X.-D. Liu, Solution of two-dimensional Riemann problems of gas dynamics by positive schemes, SIAM J. Sci. Comput. 19, 319–340 (1998). doi:10.1137/S1064827595291819
  • R. Liska and B. Wendroff, Comparison of several difference schemes on 1D and 2D test problems for the Euler equations, SIAM J. Sci. Comput. 25, 995–1017 (2003). doi:10.1137/S1064827502402120 Used in: examples/riemann_2d and examples/riemann_2d_quads (configuration 3).
  • Ames Research Staff, Equations, tables, and charts for compressible flow, NACA Report 1135 (1953). NTRS 19930091059 Used in: the oblique-shock relations against which examples/wedge and its test are checked.
  • H. Schlichting and K. Gersten, Boundary-Layer Theory, 9th ed., Springer (2017). doi:10.1007/978-3-662-52919-5 Used in: the exact viscous solutions of the tests (Couette flow, Stokes' first problem, conduction between walls).
  • V. Daru and C. Tenaud, Numerical simulation of the viscous shock tube problem by using a high resolution monotonicity-preserving scheme, Comput. Fluids 38, 664–676 (2009). doi:10.1016/j.compfluid.2008.06.008
  • G. Zhou, K. Xu and F. Liu, Grid-converged solution and analysis of the unsteady viscous flow in a two-dimensional shock tube, Phys. Fluids 30, 016102 (2018). doi:10.1063/1.4998300 Used in: examples/viscous_shock_tube and the reference wall density of tools/plot_viscous_shock_tube.py.
  • C. H. K. Williamson, Vortex dynamics in the cylinder wake, Annu. Rev. Fluid Mech. 28, 477–539 (1996). doi:10.1146/annurev.fl.28.010196.002401
  • C. Liu, X. Zheng and C. H. Sung, Preconditioned multigrid methods for unsteady incompressible flows, J. Comput. Phys. 139, 35–57 (1998). doi:10.1006/jcph.1997.5859
  • J. Park, K. Kwon and H. Choi, Numerical solutions of flow past a circular cylinder at Reynolds numbers up to 160, KSME Int. J. 12, 1200–1205 (1998). doi:10.1007/BF02942594 Used in: examples/cylinder (Strouhal number, drag and lift at Re = 100).
  • T. A. Johnson and V. C. Patel, Flow past a sphere up to a Reynolds number of 300, J. Fluid Mech. 378, 19–70 (1999). doi:10.1017/S0022112098003206
  • A. G. Tomboulides, S. A. Orszag and G. E. Karniadakis, Direct and large-eddy simulation of the flow past a sphere, in Engineering Turbulence Modelling and Experiments 2, Elsevier, 273–282 (1993). doi:10.1016/B978-0-444-89802-9.50030-7
  • A. G. Tomboulides and S. A. Orszag, Numerical investigation of transitional and weak turbulent flow past a sphere, J. Fluid Mech. 416, 45–73 (2000). doi:10.1017/S0022112000008880
  • J. Kim, D. Kim and H. Choi, An immersed-boundary finite-volume method for simulations of flow in complex geometries, J. Comput. Phys. 171, 132–150 (2001). doi:10.1006/jcph.2001.6778
  • G. S. Constantinescu and K. D. Squires, LES and DES investigations of turbulent flow over a sphere at Re = 10,000, Flow Turbul. Combust. 70, 267–298 (2003). doi:10.1023/B:APPL.0000004937.34078.71 Used in: examples/sphere_re300 and tools/plot_sphere_re300.py (Strouhal number, mean drag and lift at Re = 300: Johnson & Patel, Tomboulides et al., Kim et al., and the laminar Re = 300 validation of Constantinescu & Squires; Tomboulides & Orszag describe the regime, a single-frequency shedding that keeps a plane of symmetry).
  • M. E. Brachet, D. I. Meiron, S. A. Orszag, B. G. Nickel, R. H. Morf and U. Frisch, Small-scale structure of the Taylor–Green vortex, J. Fluid Mech. 130, 411–452 (1983). doi:10.1017/S0022112083001159
  • W. M. van Rees, A. Leonard, D. I. Pullin and P. Koumoutsakos, A comparison of vortex and pseudo-spectral methods for the simulation of periodic vortical flows at high Reynolds numbers, J. Comput. Phys. 230, 2794–2805 (2011). doi:10.1016/j.jcp.2010.11.031
  • Z. J. Wang et al., High-order CFD methods: current status and perspective, Int. J. Numer. Methods Fluids 72, 811–845 (2013). doi:10.1002/fld.3767 Used in: examples/taylor_green_3d, compared with the 512³ pseudo-spectral DNS of the International Workshop on High-Order CFD Methods (case C3.5; data).
  • E. Johnsen, J. Larsson, A. V. Bhagatwala, W. H. Cabot, P. Moin, B. J. Olson, P. S. Rawat, S. K. Shankar, B. Sjögreen, H. C. Yee, X. Zhong and S. K. Lele, Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves, J. Comput. Phys. 229, 1213–1237 (2010). doi:10.1016/j.jcp.2009.10.028
  • S. K. Lele, Simulations of turbulent flows with strong shocks and density variations, final report DE-FC02-06ER25787, Stanford University (2012). doi:10.2172/1052207
  • A. Subramaniam, M. L. Wong and S. K. Lele, A high-order weighted compact high resolution scheme with boundary closures for compressible turbulent flows with shocks, J. Comput. Phys. 397, 108822 (2019). doi:10.1016/j.jcp.2019.07.021
  • R. Samtaney, D. I. Pullin and B. Kosović, Direct numerical simulation of decaying compressible turbulence and shocklet statistics, Phys. Fluids 13, 1415–1430 (2001). doi:10.1063/1.1355682 Used in: examples/isotropic_turbulence and tools/plot_isotropic_turbulence.py: the decaying isotropic turbulence case of Johnsen et al. and their filtered 256³ reference (read from the same figure in Lele's report), the filtered 512³ DNS and spectra of Subramaniam et al. (whose appendix F gives the post-processing of both references), and the regime of Samtaney et al.; also viscosity_model = "power_law".
  • G. R. Ruetsch, L. Vervisch and A. Liñán, Effects of heat release on triple flames, Phys. Fluids 7, 1447–1454 (1995). doi:10.1063/1.868531
  • D. Veynante, L. Vervisch, T. Poinsot, A. Liñán and G. Ruetsch, Triple flame structure and diffusion flame stabilization, Proceedings of the Summer Program 1994, Center for Turbulence Research, Stanford University (1994). oa.upm.es/1537 Used in (these two): examples/triple_flame and tools/triple_flame_analysis.py (the propagation speed UF / SL against the mixing thickness at the flame, DTF, and the weak-gradient limit √(ρu / ρb)).
  • T. Poinsot, D. Veynante and S. Candel, Diagrams of premixed turbulent combustion based on direct simulation, Proc. Combust. Inst. 23, 613–619 (1990). doi:10.1016/S0082-0784(06)80308-5
  • T. Poinsot, D. Veynante and S. Candel, Quenching processes and premixed turbulent combustion diagrams, J. Fluid Mech. 228, 561–606 (1991). doi:10.1017/S0022112091002823 Used in (these two): examples/flame_vortex, examples/flame_vortex_quenching and tools/flame_vortex_analysis.py (the vortex-pair scales r and u′, the four outcomes and the 5% cut-off of the spectral diagram; their one-step flame with a linear heat loss, Le = 1.2, and its quenching).
  • Y. Ju, H. Guo, K. Maruta and F. Liu, On the extinction limit and flammability limit of non-adiabatic stretched methane–air premixed flames, J. Fluid Mech. 342, 315–334 (1997). doi:10.1017/S0022112097005636
  • Y. Ju, G. Masuya and P. D. Ronney, Effects of radiative emission and absorption on the propagation and extinction of premixed gas flames, Proc. Combust. Inst. 27, 2619–2626 (1998). doi:10.1016/S0082-0784(98)80116-1 Used in (these two): the radiative flammability limits of the 1D radiation checks (docs/design/chemistry.md): the optically thin limit flame speed near e-1/2 of the adiabatic one, and the limits of the optically thin model.
  • The spherical explosion of examples/explosion_3d is from Toro (2009), §17.1.3.
  • W. F. Noh, Errors for calculations of strong shocks using an artificial viscosity and an artificial heat flux, J. Comput. Phys. 72, 78–120 (1987). doi:10.1016/0021-9991(87)90074-X Used in: examples/noh_axisymmetric and tools/plot_noh.py (the spherical Noh problem and its exact solution).
  • R. Clift, J. R. Grace and M. E. Weber, Bubbles, Drops, and Particles, Academic Press (1978). Used in: examples/sphere_axisymmetric and tools/plot_sphere_axisymmetric.py (correlations of the drag coefficient and the separation angle of a sphere in steady flow, 20 < Re < 260), with Johnson & Patel (1999) for the wake length.
  • R. Sankaran, H. G. Im, E. R. Hawkes and J. H. Chen, The effects of non-uniform temperature distribution on the ignition of a lean homogeneous hydrogen–air mixture, Proc. Combust. Inst. 30, 875–882 (2005). doi:10.1016/j.proci.2004.08.176
  • J. H. Chen, E. R. Hawkes, R. Sankaran, S. D. Mason and H. G. Im, Direct numerical simulation of ignition front propagation in a constant volume with temperature inhomogeneities: I. Fundamental analysis and diagnostics, Combust. Flame 145, 128–144 (2006). doi:10.1016/j.combustflame.2005.09.017
  • E. R. Hawkes, R. Sankaran, P. P. Pébay and J. H. Chen, Direct numerical simulation of ignition front propagation in a constant volume with temperature inhomogeneities: II. Parametric study, Combust. Flame 145, 145–159 (2006). doi:10.1016/j.combustflame.2005.09.018
  • T. Passot and A. Pouquet, Numerical simulation of compressible homogeneous flows in the turbulent regime, J. Fluid Mech. 181, 441–466 (1987). doi:10.1017/S0022112087002167 Used in: examples/autoignition_2d (the configuration of Chen et al. and Hawkes et al., the displacement-speed diagnostic of their fronts, and the regime criterion of Sankaran et al. in tools/autoignition_analysis.py; the Passot–Pouquet spectrum of the initial temperature and velocity fields in tools/autoignition_restart.py).