The Active Flux Scheme for Nonlinear Problems

• DOI: 10.1007/s10915-020-01381-z
• Publication Date: 2020-12-21T03:02:44Z
• Authors: Wasilij Barsukow
• Keywords: Numerical Analysis (math.NA); 01 natural sciences; 1712 Software; 10123 Institute of Mathematics; 510 Mathematics; 2604 Applied Mathematics; 2200 General Engineering; FOS: Mathematics; Mathematics - Numerical Analysis; 0101 mathematics; 2614 Theoretical Computer Science; 2612 Numerical Analysis; 2605 Computational Mathematics; 35L65, 35L45, 65M08, 65M25; 1703 Computational Theory and Mathematics

AbstractThe 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.