Connections between the discontinuous Galerkin method and high-order flux reconstruction schemes

D. de Grazia, G. Mengaldo, D. Moxey, P. E. Vincent, S. J. Sherwin

Int. J. Numer. Meth. Fl., vol. 75, pp. 860–877 (2014)

@article{degrazia-2014,
  title = {{Connections between the discontinuous Galerkin method and high-order flux reconstruction schemes}},
  author = {de Grazia, D. and Mengaldo, G. and Moxey, D. and Vincent, P. E. and Sherwin, S. J.},
  volume = {75},
  number = {12},
  issn = {1097-0363},
  doi = {10.1002/fld.3915},
  pages = {860--877},
  year = {2014},
  url = {https://davidmoxey.uk/assets/pubs/2014-frdg.pdf},
  journal = ijnmf,
  abstract = {With high-order methods becoming more widely adopted throughout the field of computational fluid dynamics, the development of new computationally efficient algorithms has increased tremendously in recent years. The flux reconstruction approach allows various well-known high order schemes to be cast within a single unifying framework. Whilst a connection between flux reconstruction and the discontinuous Galerkin method has been established elsewhere, it still remains to fully investigate the explicit connections between the many popular variants of the discontinuous Galerkin method and the flux reconstruction approach. In this work, we closely examine the connections between three nodal versions of tensor product discontinuous Galerkin spectral element approximations and two types of flux reconstruction schemes for solving systems of conservation laws on quadrilateral meshes. The different types of discontinuous Galerkin approximations arise from the choice of the solution nodes of the Lagrange basis representing the solution and from the quadrature approximation used to integrate the mass matrix and the other terms of the discretisation. By considering both a linear and nonlinear advection equation on a regular grid, we examine the mathematical properties which connect these discretisations. These arguments are further confirmed by the results of an empirical numerical study.}
}

Flux reconstruction casts a number of well-known high-order schemes within one framework, and its connection to the discontinuous Galerkin method is established in outline but not for all the DG variants in use. This paper works out the connections between three nodal tensor-product discontinuous Galerkin spectral element approximations and two flux reconstruction schemes on quadrilateral meshes, where the variants differ in the choice of solution nodes and in the quadrature used.