We are interested in the accuracy of the finite volume scheme, on a LGL subcell grid induced by a LGL-DGSEM discretization. For comparison, we look at a Kelvin-Helmholtz-Instability test problem, simulated with 4² LGL nodes per element and 32² elements, i.e. 128² spatial degrees of freedom. The high-order DGSEM uses the entropy-conserving split-form powered by the flux of Chandrashekar in the volume, and the HLLC flux at the element interfaces. Positivity is controlled by adding subcell FV where the positivity is not fulfilled with the amount needed. All FV discretizations use the HLLC flux at the element interfaces and at the subcell interfaces as well.
The first results show the high-order DGSEM result, which is formally 4th order accurate for smooth problems:
The next result uses a subcell finite volume approximation on the LGL-subcell grid, directly, i.e. without spatial reconstruction (piecewise constant approximation). The result is very dissipative:
The last results show the improvement when using a piecewise linear reconstruction with monotonized-central slope limiter on the LGL-subcell grid. The reconstruction is inner-element only, as it does not use element neighbor information. Thus, at the very first and very last LGL subcell, no reconstruction is used. Still, the accuracy is recovered nicely by this simple modification of the subcell FV scheme:
