---
title: Bibliography
summary: The literature Gyre's methods come from, grouped by topic.
section: reference
order: 6
level: 2
keywords: [references, literature, bibliography]
---
<!-- gyre docs: expanded Markdown of reference/bibliography.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. -->

Bibliographic details were entered by hand; check them against the publisher before citing.

## Split-form and entropy-stable DG

* E. Tadmor (1987). The numerical viscosity of entropy stable schemes for systems of
  conservation laws. I. *Mathematics of Computation* 49, 91–103.
* T. C. Fisher, M. H. Carpenter (2013). High-order entropy stable finite difference schemes for
  nonlinear conservation laws: finite domains. *Journal of Computational Physics* 252, 518–557.
* M. H. Carpenter, T. C. Fisher, E. J. Nielsen, S. H. Frankel (2014). Entropy stable spectral
  collocation schemes for the Navier–Stokes equations: discontinuous interfaces. *SIAM Journal
  on Scientific Computing* 36(5), B835–B867.
* G. J. Gassner, A. R. Winters, D. A. Kopriva (2016). Split form nodal discontinuous Galerkin
  schemes with summation-by-parts property for the compressible Euler equations. *Journal of
  Computational Physics* 327, 39–66.
* D. A. Kopriva (2009). *Implementing Spectral Methods for Partial Differential Equations.*
  Springer.

## Two-point and interface fluxes

* F. Ismail, P. L. Roe (2009). Affordable, entropy-consistent Euler flux functions II: entropy
  production at shocks. *Journal of Computational Physics* 228, 5410–5436.
* P. Chandrashekar (2013). Kinetic energy preserving and entropy stable finite volume schemes
  for compressible Euler and Navier–Stokes equations. *Communications in Computational Physics*
  14(5), 1252–1286.
* H. Ranocha (2018). *Generalised summation-by-parts operators and entropy stability of
  numerical methods for hyperbolic balance laws.* PhD thesis, TU Braunschweig.
* C. A. Kennedy, A. Gruber (2008). Reduced aliasing formulations of the convective terms within
  the Navier–Stokes equations for a compressible fluid. *Journal of Computational Physics* 227,
  1676–1700.
* P. L. Roe (1981). Approximate Riemann solvers, parameter vectors, and difference schemes.
  *Journal of Computational Physics* 43, 357–372.
* E. F. Toro, M. Spruce, W. Speares (1994). Restoration of the contact surface in the
  HLL-Riemann solver. *Shock Waves* 4, 25–34.
* T. J. Barth (1999). Numerical methods for gasdynamic systems on unstructured meshes. In
  *An Introduction to Recent Developments in Theory and Numerics for Conservation Laws*,
  Lecture Notes in Computational Science and Engineering 5, Springer, 195–285.
* A. R. Winters, D. Derigs, G. J. Gassner, S. Walch (2017). A uniquely defined entropy stable
  matrix dissipation operator for high Mach number ideal MHD and compressible Euler
  simulations. *Journal of Computational Physics* 332, 274–289.

## Curvilinear meshes

* D. A. Kopriva (2006). Metric identities and the discontinuous spectral element method on
  curvilinear meshes. *Journal of Scientific Computing* 26(3), 301–327.
* E. Luke, E. Collins, E. Blades (2012). A fast mesh deformation method using explicit
  interpolation. *Journal of Computational Physics* 231, 586–601.

## Viscous discretization and adjoint consistency

* F. Bassi, S. Rebay (1997). A high-order accurate discontinuous finite element method for the
  numerical solution of the compressible Navier–Stokes equations. *Journal of Computational
  Physics* 131, 267–279.
* F. Bassi, S. Rebay, G. Mariotti, S. Pedinotti, M. Savini (1997). A high-order accurate
  discontinuous finite element method for inviscid and viscous turbomachinery flows. *Proc. 2nd
  European Conference on Turbomachinery Fluid Dynamics and Thermodynamics.*
* D. N. Arnold, F. Brezzi, B. Cockburn, L. D. Marini (2002). Unified analysis of discontinuous
  Galerkin methods for elliptic problems. *SIAM Journal on Numerical Analysis* 39(5), 1749–1779.
* R. Hartmann (2007). Adjoint consistency analysis of discontinuous Galerkin discretizations.
  *SIAM Journal on Numerical Analysis* 45(6), 2671–2696.
* R. Hartmann, T. Leicht (2015). Generalized adjoint consistent treatment of wall boundary
  conditions for compressible flows. *Journal of Computational Physics* 300, 754–778.
* G. J. Gassner, A. R. Winters, F. J. Hindenlang, D. A. Kopriva (2018). The BR1 scheme is stable
  for the compressible Navier–Stokes equations. *Journal of Scientific Computing* 77, 154–200.

## Turbulence modelling

* P. R. Spalart, S. R. Allmaras (1994). A one-equation turbulence model for aerodynamic flows.
  *La Recherche Aérospatiale* 1, 5–21.
* S. R. Allmaras, F. T. Johnson, P. R. Spalart (2012). Modifications and clarifications for the
  implementation of the Spalart–Allmaras turbulence model. *Seventh International Conference
  on Computational Fluid Dynamics (ICCFD7)*, paper ICCFD7-1902.
* T. A. Oliver, D. L. Darmofal (2009). Analysis of dual consistency for discontinuous Galerkin
  discretizations of source terms. *SIAM Journal on Numerical Analysis* 47(5), 3507–3525.

## Implicit solvers

* D. A. Knoll, D. E. Keyes (2004). Jacobian-free Newton–Krylov methods: a survey of approaches
  and applications. *Journal of Computational Physics* 193, 357–397.
* Y. Saad (1993). A flexible inner-outer preconditioned GMRES algorithm. *SIAM Journal on
  Scientific Computing* 14(2), 461–469.
* C. T. Kelley, D. E. Keyes (1998). Convergence analysis of pseudo-transient continuation.
  *SIAM Journal on Numerical Analysis* 35(2), 508–523.
* K. J. Fidkowski, T. A. Oliver, J. Lu, D. L. Darmofal (2005). p-Multigrid solution of high-order
  discontinuous Galerkin discretizations of the compressible Navier–Stokes equations. *Journal
  of Computational Physics* 207, 92–113.
* D. J. Mavriplis (1998). Multigrid strategies for viscous flow solvers on anisotropic
  unstructured meshes. *Journal of Computational Physics* 145, 141–165.
* M. H. Carpenter, C. A. Kennedy (1994). *Fourth-order 2N-storage Runge–Kutta schemes.* NASA
  Technical Memorandum 109112.

## Adjoints, error estimation and MDO

* R. Becker, R. Rannacher (2001). An optimal control approach to a posteriori error estimation
  in finite element methods. *Acta Numerica* 10, 1–102.
* D. A. Venditti, D. L. Darmofal (2002). Grid adaptation for functional outputs: application to
  two-dimensional inviscid flows. *Journal of Computational Physics* 176, 40–69.
* K. J. Fidkowski, D. L. Darmofal (2011). Review of output-based error estimation and mesh
  adaptation in computational fluid dynamics. *AIAA Journal* 49(4), 673–694.
* J. R. R. A. Martins, J. T. Hwang (2013). Review and unification of methods for computing
  derivatives of multidisciplinary computational models. *AIAA Journal* 51(11), 2582–2599.
* G. K. W. Kenway, G. J. Kennedy, J. R. R. A. Martins (2014). Scalable parallel approach for
  high-fidelity steady-state aeroelastic analysis and adjoint derivative computations. *AIAA
  Journal* 52(5), 935–951.
* J. S. Gray, J. T. Hwang, J. R. R. A. Martins, K. T. Moore, B. A. Naylor (2019). OpenMDAO: an
  open-source framework for multidisciplinary design, analysis, and optimization. *Structural
  and Multidisciplinary Optimization* 59, 1075–1104.
* J. R. R. A. Martins, A. Ning (2021). *Engineering Design Optimization.* Cambridge University
  Press.

## Reference data

* S. C. R. Dennis, G.-Z. Chang (1970). Numerical solutions for steady flow past a circular
  cylinder at Reynolds numbers up to 100. *Journal of Fluid Mechanics* 42(3), 471–489.
* B. Fornberg (1980). A numerical study of steady viscous flow past a circular cylinder.
  *Journal of Fluid Mechanics* 98(4), 819–855.
* NASA Langley Research Center, Turbulence Modeling Resource: 2D zero pressure gradient flat
  plate verification case, Spalart–Allmaras results (skin friction of CFL3D and FUN3D on the
  545 × 385 grid). The values used are stored with their source in
  `src/gyre/verify/studies_turbulence.py`.
* R. Clift, J. R. Grace, M. E. Weber (1978). *Bubbles, Drops, and Particles.* Academic Press
  (standard drag curve of the sphere).
* T. A. Johnson, V. C. Patel (1999). Flow past a sphere up to a Reynolds number of 300.
  *Journal of Fluid Mechanics* 378, 19–70.
* Lord Rayleigh (1916). On the flow of compressible fluid past an obstacle. *Philosophical
  Magazine* 32, 1–6 (the Janzen–Rayleigh expansion of the compressible potential in powers of
  the Mach number squared).

## Software

* J. Bradbury et al. (2018). JAX: composable transformations of Python+NumPy programs.
* Model Context Protocol specification (2024–).

## Relevant to the stated limits

* S. Hennemann, A. M. Rueda-Ramírez, F. J. Hindenlang, G. J. Gassner (2021). A provably entropy
  stable subcell shock capturing approach for high order split form DG for the compressible
  Euler equations. *Journal of Computational Physics* 426, 109935. — the natural route to shock
  capturing within this formulation.