High order positivity-preserving discontinuous Galerkin schemes for radiative transfer equations on triangular meshes

Abstract

It is an important and challenging issue for the numerical solution of radiative transfer equations to maintain both high order accuracy and positivity. For the two-dimensional radiative transfer equations, Ling et al. give a counterexample (Ling et al. (2018) [13]) showing that unmodulated discontinuous Galerkin (DG) solver based either on the Pk or Qk polynomial spaces could generate negative cell averages even if the inflow boundary value and the source term are both positive (and, for time dependent problems, also a nonnegative initial condition). Therefore the positivity-preserving frameworks in Zhang and Shu (2010) [28] and Zhang et al. (2012) [29] which are based on the value of cell averages being positive cannot be directly used to obtain a high order conservative positivity-preserving DG scheme for the radiative transfer equations neither on rectangular meshes nor on triangular meshes. In Yuan et al. (2016) [26], when the cell average of DG schemes is negative, a rotational positivity-preserving limiter is constructed which could keep high order accuracy and positivity in the one-dimensional radiative transfer equations with Pk polynomials and could be straightforwardly extended to two-dimensional radiative transfer equations on rectangular meshes with Qk polynomials (tensor product polynomials). This paper presents an extension of the idea of the above mentioned one-dimensional rotational positivity-preserving limiter algorithm to two-dimensional high order positivity-preserving DG schemes for solving steady and unsteady radiative transfer equations on triangular meshes with Pk polynomials. The extension of this method is conceptually plausible but highly nontrivial. We first focus on finding a special quadrature rule on a triangle which should satisfy some conditions. The most important one is that the quadrature points can be arranged on several line segments, on which we can use the one-dimensional rotational positivity-preserving limiter. Since the number of the quadrature points is larger than the number of basis functions of Pk polynomial space, we determine a k-th polynomial by a L2-norm Least Square subject to its cell average being equal to the weighted average of the values on the quadrature points after using the rotational positivity-preserving limiter. Since the weights used here are the quadrature weights which are positive, then the cell average of the modified polynomial is nonnegative. And the final modified polynomial can be obtained by using the two-dimensional scaling positivity-preserving limiter on the triangular element. We theoretically prove that our rotational positivity-preserving limiter on triangular meshes could keep both high order accuracy and positivity. It is relatively simple to implement, and also does not affect convergence to weak solutions. The numerical results validate the high order accuracy and the positivity-preserving properties of our schemes. The advantage of the triangular meshes on handling complex domain is also presented in our numerical examples.