Euler Equations for Ideal Gases

[ranocha]

Additional numerical fluxes:

• 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}
  }
  

• 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}
  }
  

• 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}
  }
  

• 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}
  }
  

• 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}
  }
  

• 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}
  }
  

• 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}
  }
  

• Ranocha

Additional numerical fluxes for surface terms:

• HLL: One choice of the speeds is implemented.
• 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.