Active flux scheme for time-dependent, viscous, compressible flows

The Active Flux scheme is a Finite Volume scheme with additional degrees of freedom on the cell boundaries. These additional nodes allow for third-order accuracy to be achieved, though extensions to higher orders are also possible. Unlike Godunov-type Finite Volume schemes, Active Flux makes use of a continuous reconstruction across the cell faces and does not require a Riemann solver. The variable values at the additional degrees of freedom on the cell faces are evolved independently from the cell-average values, using (non-conservative) compact-stencil schemes most suited to the mathematical and physical properties of the equations being solved. Overall conservation is enforced via the cell-average values which are updated in a step following the evolution of the additional degrees of freedom. The fundamentals of the Active Flux scheme are presented using a simple application on the Burgers equation. An evolution operator for both linear and nonlinear hyperbolic conservation systems is presented and then extended to include source terms. Example applications are made on the Euler equations and a hyperbolic formulation of the diffusion equation. Lastly, for the compressible Navier-Stokes equations, a hyperbolic formulation is presented together with an operator splitting approach. These allow for the Active Flux evolution operators to be applied to the numerical computation of viscous, compressible flows