We build on active flux (AF) schemes and extend the method to the two-dimensional linear advection, linear acoustics, and linearized Euler equations. The active flux method independently updates edge and centroid values. Because the interface fluxes are calculated from independently updated quantities, we say that they are actively updated, as opposed to a more traditional finite-volume scheme where the fluxes are updated passively from the conserved values. The one-dimensional active flux method is reformulated using Lagrange basis functions and the basic features of the method are reviewed. The two-dimensional formulation also uses standard basis functions, but includes a bubble function to maintain conservation. A novel approach for solving the linear wave system is presented that uses spherical means to compute the edge updates for the flux calculation. We demonstrate that the AF method is third-order accurate for advection, acoustics, and the linearized Euler equations.

A class of active flux schemes is developed and employed to solve model problems for the linear advection equation and non-linear Burgers equation. The active flux schemes treat the edge values, and hence the fluxes, as independent variables, doubling the degrees of freedom available to describe the solution without enlarging the stencil. Schemes up to third order accurate are explored. Oscillations generated by the higher-order schemes are controlled through the use of a characteristic-based limiter that preserves true extrema in the solution without excessive clipping.

We introduce a new formulation of active flux schemes and extend the idea to systems of hyperbolic equations. Active flux schemes treat the edge values, and hence the fluxes, as independent variables, doubling the degrees of freedom available to describe the solution without enlarging the stencil. Schemes up to third order accurate are explored. The limiter employed uses solution characteristics to set the bounds for the edge updates. The process reduces to simply accessing the solution history from memory and ensuring that the updates stay within the bounded range. The limited edge values are then used to construct the fluxes that conservatively update the centroid value. Using data that most closely follows the solution characteristics allows the limiter to better maintain true extrema in the solution. The scheme is used to generate 1-D solutions for the linear advection, Burgers', and Euler equations.

In this paper, we construct active flux schemes for 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 diffusion problems by the first-order hyperbolic system method, in which a numerical scheme is constructed based on a first-order hyperbolic system that is equivalent to the diffusion equation in the steady state. Active flux schemes are first developed for a generic hyperbolic system with source term, and then immediately applied to the hyperbolized diffusion system to generate a steady-state solver. Time-accurate schemes are constructed by implicit time integration where the steady-state solver is employed to solve the implicit residual equations. Numerical results show that third-order accuracy is obtained in both the solution and the gradient on irregular grids.

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