High order conservative positivity-preserving discontinuous Galerkin method for stationary hyperbolic equations

Abstract

This is a follow-up work of Yuan et al. (2016) [19] and Ling et al. (2018) [13] that further investigates the positivity-preserving discontinuous Galerkin (DG) methods for stationary hyperbolic equations. In 2016, Yuan et al. proposed a high order positivity-preserving DG method for stationary hyperbolic equations with constant coefficients, but the scheme has to be used in combination with a non-conservative rotational limiter introduced in case of negative cell averages. Ling et al. (2018) improved the results in one dimensional space by rigorously proving the positivity of cell averages of the unmodulated DG scheme, which allows the conservative scaling limiter in Zhang and Shu (2010) [22] to be used to maintain positivity without affecting accuracy, but extension to two space dimensions requires an augmentation of the DG space and works only in the second order case. Considering that the aforementioned works only address stationary hyperbolic equations with constant coefficients and higher than second order conservative methods are still unavailable in two and three space dimensions, we propose high order conservative positivity-preserving DG methods for variable coefficient and nonlinear stationary hyperbolic equations in one dimension and constant coefficient stationary hyperbolic equations in two and three dimensions, via a suitable quadrature in the DG framework. We show the good performance of the algorithms by ample numerical experiments, including their applications in time-dependent problems.