Sharp CFL bounds for maximum-principle-satisfying and positivity-preserving central discontinuous Galerkin methods

High-order central discontinuous Galerkin (central DG) methods evolve two polynomial solutions on overlapping primal and dual meshes and avoid Riemann solvers, but the known CFL-type conditions guaranteeing that the forward-Euler update preserves the maximum principle (scalar conservation laws) or positivity of density and pressure (compressible Euler) are sufficient-only and far more restrictive than the time steps these schemes tolerate in practice. The problem — the open core arising from the sufficient-only CFL bounds proved by Li, Li, Li and Xu (2016) and from the explicit acknowledgment in later work (Wu, Jiang and Shu, 2023) that such theoretical CFL conditions are "sufficient rather than necessary", after later work settled the unstructured-mesh and ideal-MHD extensions — is to determine the sharp, necessary and sufficient time-step conditions for bound preservation of cell averages in central DG, or to construct central DG schemes and limiters with provably larger admissible CFL numbers than the known sufficient bounds, with rigorous proofs in the original 2016 setting of scalar laws and the Euler equations on Cartesian overlapping meshes.

Background

A scalar conservation law $u_t+F(u)_x=0$ has entropy solutions that obey a strict maximum principle: $u(x,t)$ stays within $[m,M]$ , the range of the initial data. For the compressible Euler equations, physical solutions must keep density and pressure positive, i.e. the state vector must stay in the convex admissible set $H=\{U:\rho>0,\ p>0\}$ . Numerical discretizations that provably respect these bounds are prized for robustness, because overshoots or negative density/pressure destroy both physical relevance and the computation itself near strong shocks or low-density regions.

The discontinuous Galerkin (DG) method represents the solution as piecewise polynomials of degree $k$ and is $(k+1)$ -st order accurate. The central DG method is a DG variant in the central-scheme family: it evolves two copies of the numerical solution — $u_h^C$ on a primal mesh and $u_h^D$ on a dual mesh shifted by half a cell in each direction — with each copy updated using the other through an $L^2$ projection. This overlapping-mesh structure avoids Riemann solvers (the standard numerical-flux machinery that solves the local jump problem at each cell interface), which is attractive for complicated systems. A step from $t_n$ to $t_{n+1}$ involves the mesh-evolution time scale $\tau_n$ — the time scale entering the inter-mesh dissipation term $\frac{1}{\tau_n}\int (u_h^D-u_h^C)\,w\,dx$ of the central DG update, determined by the CFL condition specified in the positivity analysis and effectively the maximal time step allowed at that stage — and the ratio $\theta_n=\Delta t_n/\tau_n\in(0,1]$ , a free parameter at each time step.

Zhang and Shu established a general framework for bound-preserving high-order schemes: if, for the forward-Euler update, the cell averages remain in the admissible set whenever the point values at Legendre Gauss–Lobatto quadrature nodes (a quadrature whose nodes include the interval endpoints, with endpoint weight $\hat{\omega}_1$ ) are admissible, then a conservative linear scaling limiter that enforces exactly those nodal bounds yields a provably bound-preserving scheme, and strong-stability-preserving (SSP) Runge–Kutta methods — being convex combinations of forward-Euler steps — extend the property to high order in time. Li, Li, Li and Xu (2016) carried this framework over to central DG for scalar laws and the compressible Euler equations on Cartesian overlapping meshes, proving that cell averages stay in $[m,M]$ (or in $H$ ) under CFL-type bounds such as

\lambda_x\,a_x \le \frac{\theta_n}{2}\,\hat{\omega}_1

in one dimension and $\lambda_x a_x+\lambda_y a_y\le(\theta_n/2)\hat{\omega}_1$ in two dimensions, where $\lambda_x=\Delta t/\Delta x$ and $a_x$ is the maximum wave speed over both mesh solutions (for Euler, $|u|+c$ with sound speed $c$ ). A CFL (Courant–Friedrichs–Lewy) condition is a restriction of the time step relative to mesh size and wave speed; the constant involved determines how large a time step the theory permits.

These sufficient bounds are restrictive: $\hat{\omega}_1$ shrinks with polynomial degree ( $1/2$ for the 2-node rule used with $P^1$ , $1/6$ for the 3-node rule used with $P^2$ ) and the factor $\theta_n/2$ halves them further, so the provable steps are much smaller than both typical central DG practice and the linear-stability time-step conditions established by operator-bound analysis (Reyna and Li), which show central DG tolerates time steps comparable to or larger than standard DG. The 2016 theory was also restricted to Cartesian meshes, and rigorous multidimensional positivity analysis for harder systems such as ideal magnetohydrodynamics (MHD) was missing. Both of those directions have since been settled by later work (unstructured overlapping meshes in 2019; provably positivity-preserving central DG for multidimensional ideal MHD in 2023–2025, always again under sufficient-only CFL conditions), leaving the sharpness of the time-step restriction as the open core: is the factor $(\theta_n/2)\hat{\omega}_1$ really the admissibility frontier of the method, or can it be provably enlarged?

Problem Statement

For the central discontinuous Galerkin discretization of scalar conservation laws and the compressible Euler equations on overlapping primal/dual meshes with forward-Euler time stepping (and, by the convex-combination property of strong-stability-preserving (SSP) time integrators, its high-order-in-time extensions) equipped with bound-preserving limiters, determine the sharp time-step admissibility frontier for bound preservation of cell averages:

Establish a CFL-type condition that is both necessary and sufficient for the forward-Euler central DG update to map admissible cell averages (in $[m,M]$ for scalar laws; in the admissible set $H$ of positive density and pressure for Euler) to admissible cell averages. In particular, determine whether the known sufficient bounds, such as

\lambda_x\,a_x \le \frac{\theta_n}{2}\,\hat{\omega}_1

in one dimension and $\lambda_x a_x+\lambda_y a_y\le(\theta_n/2)\hat{\omega}_1$ in two dimensions (with $\lambda_x=\Delta t/\Delta x$ , $a_x$ the maximum wave speed over both mesh solutions, $\hat{\omega}_1$ the Gauss–Lobatto endpoint weight, and $\theta_n=\Delta t_n/\tau_n\in(0,1]$ , where $\tau_n$ is the mesh-evolution time scale of the central DG dissipation term, determined by the CFL condition specified in the positivity analysis), are sharp for the standard central DG method with the Zhang–Shu-type limiter, or whether strictly larger CFL constants are admissible.

Alternatively, construct central DG schemes, cell-average decompositions, or limiters for which bound preservation of cell averages is rigorously provable under a strictly larger CFL constant than the known sufficient bounds, in the original 2016 setting of scalar conservation laws and the compressible Euler equations on Cartesian overlapping meshes (1D and multi-D). Extensions to unstructured overlapping meshes or to other systems (e.g., ideal MHD) are valuable but are not required for resolution.

An answer must pin down the scheme, mesh pair, quadrature rules, limiter class, and the polynomial degree(s) for which the result is proved, and state the admissible CFL constant explicitly with a rigorous proof. Since $\theta_n$ is a free parameter in $(0,1]$ at each time step, the known bound is a joint condition on $(\Delta t_n,\tau_n)$ and the wave speed, and "sharp" means the sharp constant or the sharp admissible region in $(\theta_n,\lambda_x a_x)$ , either at fixed $\theta_n$ or uniformly or optimized over $\theta_n\in(0,1]$ . A claim that a proposed larger constant is inadmissible should be supported by an explicit admissible configuration — initial polynomial states on the overlapping meshes with bounded quadrature-node values — whose forward-Euler update exits the admissible set, which constitutes a finite, checkable witness.

References

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