Currently, the compiled binaries cl_get_devices.mexa64 and cl_run_kernel.mexa64 are included in matlab/. There may be several options:

• We leave it as it is.
• We use MatCL and tell the user to add this dependency to Matlab's load path (as currently described in the README).
• We ship binaries for Linux and other systems such as Windows. However, adding binaries to git is not really optimal.

Any thoughts, @Kostaszki, @philipheinisch?

Cf. ranocha/SummationByPartsOperators.jl#37, ranocha/SummationByPartsOperators.jl#38.

The surface terms for the Hall-Term need to be implemented to allow the use of non-periodic boundary conditions.

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.

Examples can be found in

• Sjögreen, Yee: High order entropy conservative central schemes for wide ranges of compressible gas dynamics and MHD flows
• Sjögreen, Yee, Kotov: Skew-Symmetric Splitting and Stability of High Order Central Schemes

It seems to be highly unusual to have e.g. I_Tech('REAL') = double and I_Tech('REAL4') = float4. Thus, a bit of code repetition can be avoided. If you agree, @Kostaszki, I can do this after most of the open PRs are (squash-) merged.

We should probably use separate files for testcase definitions so that we can easily switch between testcases for specific system of conservation laws.

We should compare both approaches. Currently, we use an array of structures. Since all read/write operations are encapsulated via inline functions, we should only have to add new functions and a define.

Up to now, we only have the possibility to use periodic boundaries everywhere or nowhere. For many tests cases such as 2D Alfvén waves, having the possibility to use periodic boundaries in z direction and other boundaries for x and y can be really helpful.

It may be better to add the possibility to use non-uniform time steps and adapt them to the current maximal velocity via the provided CFL number.

Why is this project now classified as C and not C++/OpenCL? Do we want to force classification as OpenCL?

Sometimes, secondary integral quantities such as the total kinetic energy or entropy can be of interest, e.g. in https://arxiv.org/abs/1604.06618. It would be nice to add a possibility to compute them, depending on the model and test case at hand.

As far as I know, the integral quantities of interest depend mostly on the physical model, e.g. ideal_gas_Euler or induction_equation. Should we split I_RunOps('save_integrals_over_time') into I_RunOps('save_integrals_over_time') (energy, entropy, ...) and I_RunOps('save_errors_over_time') (as it is now)? Then, I_RunOps('save_integrals_over_time') can depend on the physical model.

What do you think, @philipheinisch, @Kostaszki?

For the testcases with an analytical solution the possibility to calculate the error after every N timesteps should be added.

In order to use nonperiodic boundary conditions we need to implement surface terms for ideal MHD.

• Some test case involving the slow magnetosonic wave
• Some test case involving the fast magnetosonic wave
• Coupling of dipole and magnetosonic waves
• Magnetic field loop test
• Ideal Current Sheat

Some tests for linear waves can be found at https://www.astro.princeton.edu/~jstone/Athena/tests/linear-waves/linear-waves.html#oned-mhd. However, these aren't analytical solutions.