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Bibliography

In one minuteThe literature Gyre's methods come from, grouped by topic.

Written at L2 Practice 1196 words Markdown

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.

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