  In this paper, we develop the first entirely graphic processing unit (GPU)
based h-adaptive flux reconstruction (FR) method with linear trees. The
adaptive solver fully operates on the GPU hardware, using a linear quadtree for
two dimensional (2D) problems and a linear octree for three dimensional (3D)
problems. We articulate how to efficiently perform tree construction, 2:1
balancing, connectivity query, and how to perform adaptation for the flux
reconstruction method on the GPU hardware. As a proof of concept, we apply the
adaptive flux reconstruction method to solve the inviscid isentropic vortex
propagation problem on 2D and 3D meshes to demonstrate the efficiency of the
developed adaptive FR method on a single GPU card. Depending on the
computational domain size, acceleration of one or two orders of magnitude can
be achieved compared to uniform meshing. The total computational cost of
adaption, including tree manipulations, connectivity query and data transfer,
compared to that of the numerical solver, is insignificant. It can be less than
2% of the total wall clock time for 3D problems even if we perform adaptation
as frequent as every 10 time steps with an explicit 3-stage Runge--Kutta time
integrator.

Fluid Dynamics