Stationarity preserving nodal Finite Element methods for multi-dimensional linear hyperbolic balance laws via a Global Flux quadrature formulation

TL;DR

The paper tackles preserving stationary states in linear multi-dimensional hyperbolic balance laws by reformulating the spatial operator through a multidimensional Global Flux (GF) quadrature. The GF approach introduces integrated variables and a pressure potential $G_p$, enabling discrete equilibria characterized by sums of x- and y-dependent terms and coupling with sources, while maintaining high-order accuracy. Stabilization strategies compatible with GF, namely SU-GF and OSS-GF, are developed and analyzed alongside a Deferred Correction time integration to achieve up to order 6 in time. The authors provide a thorough consistency and well-prepared data framework, accompanied by extensive 2D numerical experiments (including Coriolis, mass source, and Stommel Gyre scenarios) that demonstrate dramatic improvements in stationary-state accuracy and long-time behavior relative to standard methods. Overall, the work delivers a robust, high-order, stationarity-preserving finite element framework with proven super-convergence at steady state and clear pathways to nonlinear extensions and DG/finite-volume variants.

Abstract

We consider linear, hyperbolic systems of balance laws in several space dimensions. They possess non-trivial steady states, which result from the equilibrium between derivatives of the unknowns in different directions, and the sources. Standard numerical methods fail to account for this equilibrium, and include stabilization that destroys it. This manifests itself in a diffusion of states that are supposed to remain stationary. We derive new stabilized high-order Finite Element methods based on a Global Flux quadrature: we reformulate the entire spatial operator as a mixed derivative of a single quantity, referred to as global flux. All spatial derivatives and the sources are thus treated simultaneously, and our methods are stationarity preserving. Additionally, when this formulation is combined with interpolation on Gauss-Lobatto nodes, the new methods are super-convergent at steady state. Formal consistency estimates, and strategies to construct well-prepared initial data are provided. The numerical results confirm the theoretical predictions, and show the tremendous benefits of the new formulation.

Figures (11)

• Figure 1: Notation of the degrees of freedom for a function $q$ in element $E_{ij}$ for $\mathbb Q^4$ elements
• Figure 2: Vortex with Coriolis force: time evolution of the norm of discrete divergence (${D}_x \otimes M_y \mathrm u + M_x \otimes {D}_y \mathrm v$ for classical methods and ${D}_x \otimes D_yI_y \mathrm u + D_x I_x \otimes {D}_y \mathrm v$ for GF simulations). Left column: $\mathbb Q^1$ on $40 \times 40$ mesh. Middle column: , $\mathbb Q^2$ on $20\times 20$ mesh. Right column: $\mathbb Q^3$ with $13\times 13$ mesh. Top: SU stabilization. Bottom: OSS stablization.
• Figure 3: Vortex with Coriolis force: simulations at time $T=100$ with $\mathbb Q^1$ elements and $40\times 40$ cells (first and second row), and with $\mathbb Q^3$ with $6 \times 6$ cells (third and last row).
• Figure 4: Vortex with Coriolis force: $\varepsilon=10^{-2}$ perturbation of the optimal equilibrium solution. Plot of $\lVert\mathbf{u}_{eq}-\mathbf{u}_p^{\text{SU}}\rVert$, with $\mathbf{u}_{eq}$ the optimized equilibrium \ref{['eq:coriolis_equilibrium']}. Top $\mathbb Q^3$ with 13 cells, bottom $\mathbb Q^3$ with 26 cells. All numerical results obtained using the SU method.
• Figure 5: Vortex with Coriolis force: $\varepsilon=10^{-6}$ perturbation of the optimal equilibrium solution. Plot of $\lVert\mathbf{u}_{eq}-\mathbf{u}_p^{\text{SU}}\rVert$, with $\mathbf{u}_{eq}$ the optimized equilibrium \ref{['eq:coriolis_equilibrium']}. Top $\mathbb Q^3$ with 13 cells, bottom $\mathbb Q^3$ with 26 cells. All numerical results obtained using the SU method.

Theorems & Definitions (13)

• Remark 1: Streamline upwinding and steady state preservation
• Remark 2: OSS and steady state preservation
• Remark 3: Symmetry of the GF quadrature
• Example 4: GF $\mathbb Q^1$ operators
• Proposition 5: Discrete steady states of the GF discretization
• proof
• Proposition 6: Discrete steady states of the GF equations and vanishing subcell integrals
• Proposition 7: Nodal super-convergence and line-by-line/row-by-row projection
• proof
• Proposition 8: Streamline Upwind stabilized GF scheme: discrete solutions and consistency