Third-order active-flux scheme for advection diffusion: Hyperbolic diffusion, boundary condition, and Newton solver

•Active flux scheme constructed for diffusion and extended to advection diffusion.•Hyperbolic method employed to simplify the discretization.•Third-order accuracy in both the solution and the gradient.•Non-uniqueness problem resolved by weak boundary condition•Newton solver developed for active-flux scheme. In this paper, we construct active flux schemes for advection diffusion. Active flux schemes are efficient third-order finite-volume-type schemes developed thus far for hyperbolic systems. This paper extends the active flux schemes to advection diffusion problems based on a first-order hyperbolic system formulation that is equivalent to the advection–diffusion equation in pseudo-steady state. An active flux scheme is first developed for a generic hyperbolic system with source terms, applied then to a hyperbolized diffusion system, extended to advection diffusion by incorporating the advective term as a source term, and enabled for unsteady problems by implicit time integration. Boundary conditions are discussed in relation to a non-uniqueness issue, and a weak boundary condition is shown to resolve the issue. Both for steady problems and for sub-iterations within unsteady problems, a globally coupled system of residual equations is solved by Newton’s method. Numerical results show that third-order accuracy is obtained in both the solution and the gradient on irregular grids with rapid convergence of Newton’s method, i.e., four or five residual evaluations are sufficient to obtain the design accuracy in both space and time.