Maximum principle preserving time implicit DGSEM for nonlinear scalar conservation laws

TL;DR

The paper addresses robust, high-order discretization of nonlinear scalar hyperbolic conservation laws in multiple space dimensions by developing maximum principle preserving and entropy-stable discontinuous Galerkin spectral element methods with implicit time stepping. It introduces both backward-Euler and space-time DGSEM formulations, augmented by local graph viscosity to enforce invariant-domain properties and discrete entropy inequalities for all admissible convex entropies. The authors prove discrete well-posedness (existence and uniqueness) of the nonlinear implicit systems under suitable viscosity bounds and demonstrate invariance and entropy properties; they also show that, in 1D, ES alone may fail to capture the entropy solution for nonconvex fluxes, while the graph-viscosity approach can recover the physical solution albeit with dissipation that can be mitigated by adaptive viscosity. Overall, the work provides a principled framework for achieving MPP and ES in implicit high-order DGSEM schemes, with practical implications for steady-state and time-resolved simulations and clear avenues for extensions to systems and limiter-based strategies.

Abstract

This work concerns the analysis of the discontinuous Galerkin spectral element method (DGSEM) with implicit time stepping for the numerical approximation of nonlinear scalar conservation laws in multiple space dimensions. We consider either the DGSEM with a backward Euler time stepping, or a space-time DGSEM discretization to remove the restriction on the time step. We design graph viscosities in space, and in time for the space-time DGSEM, to make the schemes maximum principle preserving and entropy stable for every admissible convex entropy. We also establish well-posedness of the discrete problems by showing existence and uniqueness of the solutions to the nonlinear implicit algebraic relations that need to be solved at each time step. Numerical experiments in one space dimension are presented to illustrate the properties of these schemes.

Figures (3)

• Figure 1: Inner and outer elements, $\kappa^-$ and $\kappa_e^+$, for $d=2$; definitions of traces $u_h^\pm$ on the interface $e$ and of the unit outward normal vector ${\bf n}_e^k={\bf n}_e({\bf x}_e^k)$ with ${\bf x}_e^k={\bf x}_e(\xi_k)$; positions of quadrature points in $\kappa^-$, ${\bf x}_\kappa^{ij}={\bf x}_\kappa({\boldsymbol \xi}_{ij})$, and on $e$, ${\bf x}_e^k$, for $p=3$ (bullets $\bullet$). The elements $\kappa$ are interpolated on the same grid of quadrature points as the numerical solution, i.e., ${\bf x}_\kappa({\boldsymbol \xi})=\sum_{i,j=0}^p\ell_i(\xi)\ell_j(\eta){\bf x}_\kappa^{ij}$.
• Figure 1: Inviscid Burgers' equation: solutions to problems 1 (top), 2 (middle), and 3 (bottom) defined in \ref{['tab:def_pbs']}. The $p+1=4$ DOFs per mesh element are displayed at the final time (bullets) and compared to the exact solution (lines).
• Figure 2: Buckley-Leverett equation: solutions to problems 4 (top) and 5 (bottom) defined in \ref{['tab:def_pbs']}. The $p+1=4$ DOFs per mesh element are displayed at the final time (bullets) and compared to the exact solution (lines).

Theorems & Definitions (24)

• Lemma 2.1
• Proof 1
• Lemma 2.2
• Proof 2
• Remark 2.3: square entropy
• Lemma 3.1
• Proof 3
• Theorem 3.2
• Proof 4
• Corollary 3.3