Abstract
The Active Flux scheme is a finite volume scheme with additional point values distributed along the cell boundary. It is third order accurate and does not require a Riemann solver. Instead, given a reconstruction, the initial value problem at the location of the point value is solved. The intercell flux is then obtained from the evolved values along the cell boundary by quadrature. Whereas for linear problems an exact evolution operator is available, for nonlinear problems one needs to resort to approximate evolution operators. This paper presents such approximate operators for nonlinear hyperbolic systems in one dimension and nonlinear scalar equations in multiple spatial dimensions. They are obtained by estimating the wave speeds to sufficient order of accuracy. Additionally, an entropy fix is introduced and a new limiting strategy is proposed. The abilities of the scheme are assessed on a variety of smooth and discontinuous setups.
Highlights
Hyperbolic m × m systems of conservation laws in d spatial dimensions have the form∂t q + ∇ · f(q) = 0 q : R+0 × Rd → Rm (1.1)The function f is called the flux
In order to evolve the cell average, finite volume methods require the knowledge of the flux through the intercell boundary
The Active Flux scheme is a finite volume scheme with additional pointwise degrees of freedom located at the cell boundary
Summary
Hyperbolic m × m systems of conservation laws in d spatial dimensions have the form. The function f is called the flux. Exact solutions of these equations are unavailable in general, and one needs to resort to numerical methods. Cell based methods consider the computational domain to be partitioned into cells. A certain number of discrete degrees of freedom are associated with every cell: e.g. finite.
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