Active Flux and Discontinuous Galerkin look like different methods. Active Flux uses globally continuous approximations with point values at cell interfaces. Discontinuous Galerkin allows discontinuities between cells and uses polynomial bases within each cell. Different philosophies, different implementations, different communities.
Barsukow, Klingenberg, and Krotsch prove they are the same method.
For linear problems in one and multiple dimensions, a mapping exists between the degrees of freedom of Active Flux and DG such that the updates agree exactly. The two methods, derived independently from different principles, produce identical numerical solutions. The mapping extends, in some sense, to nonlinear problems.
The equivalence reveals structure that neither method shows alone. Radau polynomials and their zeros appear naturally in the AF-DG mapping — objects well-known in DG theory but with no obvious role in Active Flux. Their appearance explains DG superconvergence: the phenomenon where DG solutions converge faster at certain points than the global rate would suggest. Active Flux “functions as the underlying high-order method within DG” — the superconvergence points are where the Active Flux structure shows through the DG formulation.
Active Flux is also more economical. With K+1 degrees of freedom per cell on average, it achieves polynomial degree K+1, while DG uses the same number for degree K. The continuous interface values in Active Flux are shared between cells; DG's discontinuous formulation doubles them. Same computation, higher approximation order.
Two communities building separate methods were building the same method from different directions. The equivalence was always there, hidden by the difference in formulation.