On the conservation property of positivity-preserving discontinuous Galerkin methods for stationary hyperbolic equations

Recently, there has been a series of works on the positivity-preserving discontinuous Galerkin methods for stationary hyperbolic equations, where the notion of mass conser-vation follows from a straightforward analogy of that of time-dependent problems, i.e. conserving the mass = preserving cell averages during limiting. Based on such a notion, the implementations and theoretical proofs of positivity-preserving limited methods for stationary equations are unnecessarily complicated and constrained. As will be shown in this paper, in some extreme cases, their convergence could even be problematic. In this work, we clarify a more appropriate definition of mass conservation for limiters applied to stationary hyperbolic equations and establish the genuinely conservative high-order positivity-preserving limited discontinuous Galerkin methods based on this definition. The new methods are able to preserve the positivity of solutions of scalar linear equations and scalar nonlinear equations with invariant wind direction, with much simpler implemen-tations and easier proofs for accuracy and the Lax-Wendroff theorem, compared with the existing methods. Two types of positivity-preserving limiters preserving the local mass of stationary equations are developed to accommodate for the new definition of conservation and their accuracy are investigated. We would like to emphasize that a major advantage of the original DG scheme presented in [24] is a sweeping procedure, which allows for the computation of conservative steady-state solutions explicitly, cell by cell, without iterations, even for nonlinear equations as long as the wind direction is fixed. The main contribution of this paper is to introduce a limiting procedure to enforce positivity without changing the conservative property of this original DG scheme. The good performance of the algorithms for stationary hyperbolic equations and their applications in time-dependent problems are demonstrated by ample numerical tests.& COPY; 2023 Elsevier Inc. All rights reserved.