3DFLUX: A high-order fully three-dimensional flux integral solver for the scalar transport equation

We present a detailed derivation of a high-order, fully three-dimensional, conservative, monotonicity preserving, flux integral method for the solution of the scalar transport equation. This algorithm, named 3DFLUX, produces highly accurate solutions that are nearly unaffected by numerical dissipation, at a realistic computational cost. The performance of 3DFLUX is characterized by means of several challenging multidimensional tests. 3DFLUX is nominally third-order in space and second-order in time, however, at low Courant numbers, it appears to be superconvergent and, depending on the problem solved, is fourth-order or higher in space. Finally, 3DFLUX is used to simulate advection-diffusion of a complex temperature field in an incompressible turbulent flow of practical relevance, and its results are in excellent agreement with experimental measurements.