Euler Equations for Ideal Gases

[ranocha]

Additional numerical fluxes:

• [x] Ducros et al.

  @article{ducros2000high,
  title={High-Order Fluxes for Conservative Skew-Symmetric-Like Schemes in
        Structured Meshes: {A}pplication to Compressible Flows},
  author={Ducros, F and Laporte, F and Souleres, T and Guinot, V and Moinat, P
          and Caruelle, B},
  journal={Journal of Computational Physics},
  volume={161},
  number={1},
  pages={114--139},
  year={2000},
  publisher={Elsevier},
  doi={10.1006/jcph.2000.6492}
  }

• [x] Morinishi

  @article{morinishi2010skew,
  title={Skew-symmetric form of convective terms and fully conservative finite
        difference schemes for variable density low-{M}ach number flows},
  author={Morinishi, Yohei},
  journal={Journal of Computational Physics},
  volume={229},
  number={2},
  pages={276--300},
  year={2010},
  publisher={Elsevier},
  doi={10.1016/j.jcp.2009.09.021}
  }

• [x] Kennedy and Gruber

  @article{kennedy2008reduced,
  title={Reduced aliasing formulations of the convective terms within the
        {N}avier--{S}tokes equations for a compressible fluid},
  author={Kennedy, Christopher A and Gruber, Andrea},
  journal={Journal of Computational Physics},
  volume={227},
  number={3},
  pages={1676--1700},
  year={2008},
  publisher={Elsevier},
  doi={10.1016/j.jcp.2007.09.020}
  }

• [x] Pirozzoli

  @article{pirozzoli2011numerical,
  title={Numerical Methods for High-Speed Flows},
  author={Pirozzoli, Sergio},
  journal={Annual Review of Fluid Mechanics},
  volume={43},
  pages={163--194},
  year={2011},
  publisher={Annual Reviews},
  doi={10.1146/annurev-fluid-122109-160718}
  }

• [x] Kuya, Totani, and Kawai

  @article{kuya2018kinetic,
  title={Kinetic energy and entropy preserving schemes for compressible flows
        by split convective forms},
  author={Kuya, Yuichi and Totani, Kosuke and Kawai, Soshi},
  journal={Journal of Computational Physics},
  volume={375},
  pages={823--853},
  year={2018},
  publisher={Elsevier},
  doi={10.1016/j.jcp.2018.08.058}
  }

• [x] Ismail and Roe

  @article{ismail2009affordable,
  title={Affordable, entropy-consistent {E}uler flux functions {II}:
        {E}ntropy production at shocks},
  author={Ismail, Farzad and Roe, Philip L},
  journal={Journal of Computational Physics},
  volume={228},
  number={15},
  pages={5410--5436},
  year={2009},
  publisher={Elsevier},
  doi={10.1016/j.jcp.2009.04.021}
  }

• [x] Chandrashekar

  @article{chandrashekar2013kinetic,
  title={Kinetic Energy Preserving and Entropy Stable Finite Volume Schemes for
        Compressible {E}uler and {N}avier-{S}tokes Equations},
  author={Chandrashekar, Praveen},
  journal={Communications in Computational Physics},
  volume={14},
  number={5},
  pages={1252--1286},
  year={2013},
  doi={10.4208/cicp.170712.010313a}
  }

• [x] Ranocha

Additional numerical fluxes for surface terms:

• [ ] HLL: One choice of the speeds is implemented.
• [x] Suliciu

See e.g.

• Gassner, Winters, Kopriva: Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations
• Ranocha: Generalised Summation-by-Parts Operators and Entropy Stability of Numerical Methods for Hyperbolic Balance Laws

Additional test cases:

• [ ] Some form of an isentropic vortex

Since the Euler equations are one of the basic examples of hyperbolic conservation laws, we should probably add the to this repository.

[ranocha]

A first setup is in #16, including some numerical volume fluxes.

[ranocha]

Additionally, boundary conditions based on numerical fluxes (e.g. Suliciu, HLL) should be added.