Finite element-based invariant-domain preserving approximation of hyperbolic systems: Beyond second-order accuracy in space

Abstract

This paper proposes an invariant-domain preserving approximation technique for nonlinear conservation systems that is high-order accurate in space and time. The algorithm mixes a high-order finite element method with an invariant-domain preserving low-order method that uses the closest neighbor stencil. The construction of the flux of the low-order method is based on an idea from Abgrall et al. (2017). The mass flux of the low-order and the high-order methods are identical on each finite element cell. This allows for mass preserving and invariant-domain preserving limiting.