Bound-preserving flux limiting schemes for DG

Many mathematical models of computational fluid dynamics involve transport of conserved quantities which must lie in a certain range to be physically meaningful. The analytical or numerical solution u of a scalar conservation law is said to satisfy a maximum principle (MP) if global bounds u∗ and u ∗ exist such that u∗ ≤ u ≤ u∗ holds in the domain of definition. To enforce such inequality constraints at least for element averages in the context of discontinuous Galerkin (DG) methods, the numerical fluxes must be defined and constrained in an appropriate manner. In this work, we introduce a general framework for calculating fluxes that produce non-oscillatory DG approximations and preserve relevant global bounds for element averages even if the exact solution of the PDE violates them due to modeling errors. The proposed methodology is based on a combination of flux and slope limiting. The (optional) slope limiter adjusts the gradients to impose local bounds on pointwise values of the high-order DG solution which is used to calculate the fluxes. The flux limiter constrains changes of element averages so as to prevent violations of global bounds. Since manipulations of the target flux may introduce a consistency error, it is essential to guarantee that physically admissible fluxes remain unchanged. We propose two kinds of flux limiters which meet this requirement. The first one is of monolithic type and and its time-implicit version requires the iterative solution of a nonlinear problem. Only a fully converged solution is provably bound-preserving. The explicit version of this limiter is subject to a time step restriction which we derive in this article. The second limiter is an iterative version of the multidimensional fluxcorrected transport (FCT) algorithm. Our fractional step limiting (FSL) approach guarantees the MP property for each iterate but avoidable consistency errors may occur if the iterative process is terminated too early. Practical applicability of the proposed iterative limiters is demonstrated by numerical studies for the advection equation (hyperbolic, linear) and the Cahn–Hilliard equation (parabolic, nonlinear).