Active Flux Schemes for Systems

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.

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