A collocation scheme that is equivalent to discontinuous Galerkin discretizations
Provides theoretical equivalence between two popular numerical methods for PDEs, clarifying their relationship and guiding practical implementation.
This work proves that a spectral collocation operator with summation-by-parts property yields solutions equivalent to discontinuous Galerkin semi-discretizations using the same quadrature, regardless of degrees of freedom or quadrature positivity. Equivalence holds for linear advection and Burgers' equation, but breaks down for entropy-stable skew-symmetric splittings unless projection is used.
A spectral collocation operator with the summation-by-parts property was introduced by Chan to develop entropy-stable discontinuous Galerkin (DG) semi-discretizations (https://doi.org/10.1016/j.jcp.2018.02.033). The present work shows that semi-discretizations based on this collocation operator produce solutions that are equivalent to solutions of a DG semi-discretization using the same underlying quadrature. The equivalence holds regardless of the number of degrees of freedom in the collocation scheme and when the quadrature is not strictly positive. Extraneous degrees of freedom in the collocation scheme are associated with the nullspace of the operator and remain zero throughout an unsteady simulation. If necessary, nullspace consistency can be recovered by introducing projection-based numerical dissipation that targets only the extraneous modes. The equivalence between collocation and DG solutions is verified for the constant-coefficient advection equation and Burgers' equation on triangular meshes. The numerical results show that equivalence breaks down for entropy-stable semi-discretizations of Burgers' equation based on a skew-symmetric splitting, but that equivalence can be recovered by projecting the collocation scheme's residual onto the relevant polynomial space. In addition to investigating equivalence, the results demonstrate that the collocation operator produces semi-discretizations with favorable spectral radii compared with a commonly used summation-by-parts operator construction.