[Paper Review] Monolithic Convex Limiting for Legendre-Gauss-Lobatto Discontinuous Galerkin Spectral Element Methods

Andrés M. Rueda-Ramírez, Benjamin Bolm+2 more|arXiv (Cornell University)|Mar 1, 2023

Advanced Numerical Methods in Computational Mathematics4 citations

TL;DR

This paper introduces a monolithic convex limiting (MCL) framework for Legendre–Gauss–Lobatto discontinuous Galerkin spectral element methods (LGL-DGSEM), leveraging the inherent subcell flux structure and summation-by-parts properties of LGL nodes to enforce invariant domain preservation and entropy stability. The method achieves time-step-independent stabilization, enabling robust simulations of compressible Euler flows with strong shocks, steep gradients, and vortex-dominated regimes without artificial dissipation dependence on time step.

ABSTRACT

We extend the monolithic convex limiting (MCL) methodology to nodal discontinuous Galerkin spectral element methods (DGSEM). The use of Legendre-Gauss-Lobatto (LGL) quadrature endows collocated DGSEM space discretizations of nonlinear hyperbolic problems with properties that greatly simplify the design of invariant domain preserving high-resolution schemes. Compared to many other continuous and discontinuous Galerkin method variants, a particular advantage of the LGL spectral operator is the availability of a natural decomposition into a compatible subcell flux discretization. Representing a high-order spatial semi-discretization in terms of intermediate states, we perform flux limiting in a manner that keeps these states and the results of Runge-Kutta stages in convex invariant domains. Additionally, local bounds may be imposed on scalar quantities of interest. In contrast to limiting approaches based on predictor-corrector algorithms, our MCL procedure for LGL-DGSEM yields nonlinear flux approximations that are independent of the time-step size and can be further modified to enforce entropy stability. To demonstrate the robustness of MCL/DGSEM schemes for the compressible Euler equations, we run simulations for challenging setups featuring strong shocks, steep density gradients and vortex dominated flows.

Motivation & Objective

• To extend monolithic convex limiting (MCL) to nodal discontinuous Galerkin spectral element methods using Legendre–Gauss–Lobatto (LGL) quadrature.
• To exploit the subcell flux decomposition and diagonal mass matrices in LGL-DGSEM to simplify the design of high-resolution, invariant domain preserving schemes.
• To develop a flux-limiting strategy that maintains convex invariant domains across Runge–Kutta stages and allows local bounds on scalar quantities.
• To achieve entropy stability through MCL-based flux modifications independent of time-step size.
• To demonstrate robustness on challenging hyperbolic problems involving shocks, density gradients, and turbulent flows.

Proposed method

• Represents the high-order spatial semi-discretization in terms of intermediate states to enable flux limiting while preserving convex invariant domains.
• Uses the natural subcell flux decomposition inherent in LGL-DGSEM to define anti-diffusive corrections without artificial reconstruction.
• Implements a monolithic approach where limited fluxes are incorporated directly into the residual, avoiding predictor-corrector time-splitting.
• Enforces invariant domain preservation by constraining individual fluxes rather than flux sums, ensuring local and global bounds.
• Applies entropy stability via limiter-based fixes that maintain semi-discrete entropy inequalities.
• Employs diagonal mass matrices and summation-by-parts (SBP) operators in LGL-DGSEM to support entropy stability and high-order accuracy.

Experimental results

Research questions

• RQ1 Can monolithic convex limiting be effectively extended to LGL-DGSEM to preserve invariant domains and entropy stability?
• RQ2 How does MCL in LGL-DGSEM compare to traditional predictor-corrector FCT schemes in terms of time-step dependence and dissipation?
• RQ3 What is the performance of MCL/LGL-DGSEM on benchmark problems with strong shocks, steep gradients, and vortex-dominated flows?
• RQ4 Can local bounds on scalar quantities be enforced within the MCL framework without compromising stability?
• RQ5 Does the monolithic formulation ensure time-step-independent stabilization in nonlinear hyperbolic systems?

Key findings

• The MCL/LGL-DGSEM framework achieves time-step-independent stabilization, enabling expected time convergence in problems requiring numerical stabilization.
• The method successfully captures strong shocks, steep density gradients, and vortex-dominated flows without spurious oscillations or domain violations.
• Compared to predictor-corrector FCT schemes, MCL reduces dissipation dependence on time-step size, improving temporal convergence behavior.
• The use of LGL nodes enables a natural subcell flux decomposition and diagonal mass matrices, simplifying the implementation of convex limiting.
• Entropy stability is enforced via MCL-based flux modifications, and semi-discrete entropy inequalities are preserved through the limiting strategy.
• The implementation in Trixi.jl is reproducible and publicly available, supporting full code and result replication.