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Positivity-Preserving and Entropy-Stable Oscillation-Eliminating DGSEM for the Compressible Euler Equations on Curvilinear Meshes with Adaptive Mesh Refinement

This paper presents an extension of the entropy-stable, oscillation-eliminating discontinuous Galerkin spectral element method to adaptive mesh refinement on curvilinear meshes, introducing specialized numerical fluxes and a positivity-preserving framework to ensure high-order accuracy, global conservation, and robustness for the compressible Euler equations.

Original authors: Jieling Yang, Guosheng Fu

Published 2026-04-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jieling Yang, Guosheng Fu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to simulate the weather, or how air flows around a supersonic jet, using a computer. To do this accurately, you need to break the sky (or the space around the jet) into millions of tiny puzzle pieces called a mesh.

The problem is, the air doesn't always behave nicely. Sometimes it flows smoothly, but other times it hits a wall and creates a massive, chaotic explosion (a shockwave). If your puzzle pieces are all the same size, you either waste a lot of computer power on empty sky, or you miss the details of the explosion because your pieces are too big.

This paper presents a new, super-smart way to solve these problems using a method called Adaptive Mesh Refinement (AMR). Think of it like a camera that automatically zooms in with high resolution only where the action is happening (like a shockwave) and zooms out to a low resolution where nothing is happening.

Here is the breakdown of what the authors did, explained through simple analogies:

1. The Curved Puzzle Pieces

Most computer simulations use a grid of perfect squares. But the real world has curves (like the wing of a plane or the surface of a planet).

The Analogy: Imagine trying to tile a curved wall with square tiles. It looks ugly and leaves gaps. Instead, the authors use curvilinear meshes, which are like flexible, stretchy puzzle pieces that can bend to fit the shape of the object perfectly.

2. The "Hanging Node" Problem

When you zoom in (refine the mesh) in one area, you get small puzzle pieces next to big ones.

The Analogy: Imagine a large square tile next to four tiny square tiles. The edge of the big tile doesn't line up perfectly with the edges of the tiny ones. In math terms, these are called "hanging nodes."
The Challenge: When air flows across this mismatched boundary, the computer has to guess how to pass the information from the big tile to the small ones. If it guesses wrong, the simulation can crash (the air pressure might turn negative, which is physically impossible) or become unstable.

3. The Two New "Bridge" Strategies

The authors built two different "bridges" to connect these mismatched tiles. They found a trade-off between Safety and Precision.

Strategy A: The "Safety-First" Bridge (Entropy-Stable Flux)

How it works: This bridge is designed to be mathematically "safe." It guarantees that the total energy of the system behaves correctly and that the simulation won't blow up. It's like a sturdy, reinforced bridge that can handle any storm.
The Catch: Because it's so focused on safety, it loses a bit of sharpness. For high-level simulations (using complex math), it becomes a bit "blurry." It's like looking at a high-definition photo through a slightly foggy window. It's safe, but not perfectly accurate.

Strategy B: The "Precision-First" Bridge (Mortar-Based Flux)

How it works: This bridge is designed to be incredibly sharp. It interpolates (guesses) the data so perfectly that it keeps the high-definition quality of the simulation. It's like a high-speed fiber-optic cable.
The Catch: It doesn't have the same built-in "safety net" for energy conservation as the first bridge. However, the authors proved that if you add a specific "guardrail" (a positivity limiter), it still won't crash.
The Result: This is the winner for most practical uses because it gives you the best of both worlds: high accuracy and safety (as long as you use the guardrail).

4. The "Guardrail" (Positivity Preservation)

In fluid dynamics, things like air density and pressure must always be positive numbers. You can't have "negative air."

The Analogy: Imagine a car driving on a road. If the car goes too fast, it might fly off the road. The Zhang-Shu limiter is like a smart guardrail. If the computer calculation starts to drift toward "negative air" (a crash), the guardrail gently nudges the numbers back to safe, positive values without ruining the speed of the car.
The Breakthrough: The authors proved that their new "Precision Bridge" works perfectly with this guardrail, even on those tricky mismatched (curved) grids.

5. The Oscillation Eliminator (OE)

Sometimes, when a computer simulates a sharp shockwave, it creates weird, fake ripples (oscillations) that look like static on an old TV.

The Analogy: Think of this as a "noise-canceling" feature. The authors added a step that detects these fake ripples and smooths them out instantly, but only in the messy areas. In the smooth areas, it leaves the data alone so the picture stays crisp.

Summary: Why This Matters

Before this paper, scientists had to choose between:

Safety: A method that wouldn't crash but was blurry.
Accuracy: A method that was sharp but risky to use on complex, curved shapes.

This paper provides a toolkit that allows scientists to:

Use curved, flexible puzzle pieces to model real-world shapes.
Zoom in and out automatically to save computer power.
Use a high-precision bridge that stays sharp.
Keep a guardrail to ensure the simulation never crashes due to impossible physics.

The result is a simulation tool that is robust (won't crash), efficient (uses less computer power), and accurate (shows fine details), making it possible to model everything from supersonic jets to astrophysical explosions with unprecedented clarity.

1. Problem Statement

The paper addresses the challenge of simulating compressible flows governed by the Euler equations on curvilinear meshes using Adaptive Mesh Refinement (AMR) with nonconforming interfaces (hanging nodes).

While high-order Discontinuous Galerkin Spectral Element Methods (DGSEM) are robust for complex geometries, extending them to AMR grids introduces significant difficulties:

Nonconforming Interfaces: Standard numerical fluxes used on conforming meshes fail on interfaces where element sizes differ (e.g., a 2:1 refinement ratio).
Stability vs. Accuracy Trade-off: Existing entropy-stable formulations on nonconforming meshes often suffer from a loss of formal high-order accuracy due to negative entries in interpolation operators.
Positivity Preservation: Ensuring that physical quantities (density and pressure) remain positive is critical for stability, especially in AMR where data transfer between mesh levels can introduce non-physical states.
Oscillations: High-order methods generate spurious oscillations near shocks, requiring oscillation-eliminating (OE) procedures that must function correctly on nonconforming grids.

The authors aim to extend their previous Entropy-Stable Oscillation-Eliminating DGSEM (ES-OEDG) framework to AMR grids while maintaining entropy stability, positivity preservation, and high-order accuracy.

2. Methodology

The proposed framework is built upon a 2D curvilinear quadrilateral mesh with a 2:1 refinement constraint (allowing one level of hanging nodes). The methodology consists of four core components:

A. Entropy-Stable Interface Flux (The "ES" Flux)

The authors first construct a numerical flux for nonconforming interfaces that guarantees global conservation and a semi-discrete entropy inequality.

Construction: They use interpolation operators ( $P_{C \to F}$ ) to map coarse-side states to fine-side nodes. The flux is modified with a sign function to ensure the dissipative jump terms have the correct sign for entropy stability.
Limitation: The analysis reveals a fundamental trade-off. For polynomial degrees $N \ge 2$ , the interpolation matrices contain negative entries. To maintain entropy stability, the flux must use absolute values of these entries, which destroys the cancellation properties required for formal high-order consistency. Consequently, this flux is at most first-order accurate for $N \ge 2$ .

B. Mortar-Based Interface Flux (The "Mortar" Flux)

To recover high-order accuracy, the authors propose a classical mortar-based approach where the fine edges serve as mortars.

Construction: The solution is interpolated from the coarse edge to the fine nodes (at the solution level, not the flux level). Standard two-point local Lax–Friedrichs fluxes are then evaluated at the fine nodes.
Properties: This approach preserves formal high-order accuracy and positivity. However, it loses provable entropy stability because the interpolation at the solution level (rather than the flux level) prevents the construction of a discrete entropy inequality when negative interpolation coefficients exist.

C. Positivity Preservation

The paper rigorously extends the Zhang–Shu positivity-preserving framework to curvilinear AMR meshes.

Mechanism: They prove that under a Forward Euler time step and a suitable CFL condition, the cell-average density and pressure remain positive for both the ES flux and the Mortar flux.
Limiter: A Zhang–Shu scaling limiter is applied element-wise to ensure positivity at all nodal points while preserving the cell average.
Data Transfer: A conservative and positivity-preserving data transfer procedure is developed for AMR refinement (polynomial restriction) and coarsening ( $L^2$ projection followed by limiting).

D. Oscillation Elimination (OE) and AMR Strategy

OE Procedure: An element-local post-processing step suppresses spurious oscillations near shocks. On nonconforming meshes, jump indicators are computed by transferring traces to a common nodal set before evaluation.
AMR Loop: The algorithm uses a shock indicator to drive refinement and coarsening. The mesh is maintained under a 2:1 balance constraint. The full time-step loop includes: Time Integration $\to$ OE $\to$ Positivity Limiting $\to$ (Periodic) Mesh Adaptation.

3. Key Contributions

Theoretical Analysis of Interface Fluxes: The paper provides a rigorous proof that for $N \ge 2$ , there is an inherent trade-off between provable entropy stability and formal high-order accuracy on nonconforming interfaces due to the sign structure of interpolation operators.
Dual-Flux Framework:
- An Entropy-Stable Flux that guarantees stability but sacrifices accuracy for $N \ge 2$ .
- A Mortar-Based Flux that guarantees high-order accuracy and positivity, but lacks a proof of entropy stability.
Extension of Positivity Preservation: A rigorous proof that positivity of cell averages is preserved on curvilinear AMR meshes for both flux types, enabling the use of the Zhang–Shu limiter to ensure nodal positivity.
Complete AMR Algorithm: A robust implementation integrating shock detection, selective oscillation elimination, conservative data transfer, and positivity limiting specifically designed for curvilinear nonconforming meshes.

4. Numerical Results

The authors validate the method using the open-source MFEM framework on several benchmarks:

Isentropic Vortex (Smooth Flow):
- Confirmed that the ES flux degrades to $\approx 1.0\text{--}1.5$ order accuracy for $N=2,3$ on nonconforming meshes.
- Confirmed that the Mortar flux achieves the design order of accuracy ( $N$ ) for all polynomial degrees.
Double Mach Reflection (Strong Shocks):
- Demonstrated that the Mortar flux ( $N=2$ ) with AMR captures fine-scale structures (slip lines, downstream vortices) significantly better than a conforming mesh of similar resolution or the low-order ES flux.
High Mach Number Astrophysical Jet (Positivity Test):
- Simulated a Mach 2350 jet with extreme density/pressure contrasts. The method remained robust without failure, verifying the effectiveness of the positivity-preserving limiter on AMR meshes.
Supersonic Flow Past a Cylinder (Curvilinear AMR):
- Showed the ability of the curvilinear AMR strategy to resolve Kelvin–Helmholtz instabilities and bow shocks with moderate degrees of freedom, outperforming uniform meshes.

5. Significance and Conclusion

This work represents a significant step forward in high-order CFD for complex geometries. It provides a practical, robust algorithm for compressible flows on adaptive curvilinear meshes.

Practical Utility: The Mortar-based flux is identified as the preferred choice for high-order simulations ( $N \ge 2$ ) because it retains design accuracy and positivity, which are critical for resolving complex flow features.
Theoretical Insight: The paper clarifies the limitations of entropy-stable formulations on nonconforming grids, highlighting that achieving entropy stability, positivity, and high-order accuracy simultaneously for $N \ge 2$ remains an open challenge.
Robustness: The integration of oscillation elimination and positivity limiting ensures the method can handle extreme conditions (strong shocks, low density) without crashing, making it suitable for astrophysical and aerodynamic applications.

In summary, the authors have successfully extended a state-of-the-art DGSEM framework to AMR, providing a complete, tested, and theoretically grounded toolkit for simulating compressible flows on complex, adaptive grids.