🔢 mathematics

Convergence analysis for a finite-volume scheme for the Euler- and Navier-Stokes-Korteweg system via energy-variational solutions

This paper proves that a structure-preserving finite-volume scheme for the Euler- and Navier-Stokes-Korteweg systems converges to energy-variational solutions under mesh refinement by establishing uniform estimates and leveraging the stability of the newly extended energy-variational formulation.

Original authors: Thomas Eiter, Jan Giesselmann, Robert Lasarzik, Philipp Öffner, Robert Sauerborn

Published 2026-04-01

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Thomas Eiter, Jan Giesselmann, Robert Lasarzik, Philipp Öffner, Robert Sauerborn

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 predict the weather, or how a drop of ink spreads in a glass of water. To do this, scientists use complex mathematical equations (like the Euler and Navier-Stokes equations) that describe how fluids move. However, these equations are notoriously difficult. Sometimes, they have multiple possible answers for the same starting point, or they break down entirely when things get chaotic (like when a shockwave forms).

This paper is about building a digital simulation (a computer model) to solve these equations, specifically for fluids that have a special "surface tension" property called capillarity (think of how water beads up on a leaf). The authors, Thomas Eiter and his team, have created a new way to prove that their computer simulation is actually working correctly and converging to the "true" physical reality, even when the math gets messy.

Here is a breakdown of their work using simple analogies:

1. The Problem: The "Foggy" Fluid

Imagine trying to track a cloud of gas. In the real world, the gas might have tiny ripples or surface tension effects (like the skin on a soap bubble). The standard math models struggle here.

The Issue: When you try to solve these equations on a computer, you have to chop the space into tiny little boxes (like a grid on a graph paper). As you make the boxes smaller and smaller to get a better picture, the computer calculations can sometimes go haywire, or you might not know if the answer you get is the real answer or just a glitch.
The "Weak" Solution: In math, when a perfect, smooth answer doesn't exist, we accept a "weak" solution. Think of it like looking at a low-resolution photo. You can't see the individual pixels clearly, but you can still tell it's a face. The problem is, sometimes there are too many different "low-res" faces that could fit the data.

2. The Solution: The "Energy Variational" Safety Net

The authors introduce a new concept called Energy-Variational Solutions.

The Analogy: Imagine you are trying to balance a stack of books. You know the total weight (Energy) of the stack. Even if the books wobble and shift (chaos), the total weight stays the same or decreases (due to friction).
The Innovation: Instead of trying to force the computer to find a perfect, smooth path for every single molecule, the authors say: "Let's just track the total energy and make sure the computer simulation never creates energy out of thin air."
They created a "Safety Net" (a mathematical inequality) that says: As long as the simulation respects the laws of energy conservation and dissipation, it is a valid solution. This allows them to prove that their computer model is stable and reliable, even if the fluid is doing something wild.

3. The Method: The "Structure-Preserving" Lego Set

The authors used a Finite-Volume Scheme.

The Analogy: Imagine you are building a model of a river using Lego bricks. A standard Lego set might let you build a river that magically gains water out of nowhere or loses water.
The Fix: The authors built a special "Structure-Preserving" Lego set. This set has a rule: You cannot add or remove water (mass) or momentum (movement) between the bricks.
They also added a "friction" rule (viscosity) and a "surface tension" rule (Korteweg stress) directly into the way the bricks connect. Because the Lego set follows the exact same physical rules as the real river, the simulation stays honest.

4. The Proof: The "Zoom-In" Test

The core of the paper is the Convergence Analysis.

The Experiment: They ran their simulation with big blocks (low resolution), then medium blocks, then tiny blocks (high resolution).
The Result: They proved mathematically that as the blocks get infinitely small (zooming in to the limit), the simulation doesn't just get "better"; it settles down into a specific, unique pattern that matches their "Energy-Variational" safety net.
Why it matters: Before this, people worried that as you zoomed in, the simulation might jump between different chaotic answers. This paper proves that the simulation is stable and will always point toward the same physical truth, defined by how energy behaves.

5. Why This is a Big Deal

No "Strong" Solution Needed: Usually, to prove a computer model works, you need to know the "perfect" answer exists first. But for these complex fluids, the perfect answer might not exist! This paper proves the computer model works without needing to know the perfect answer beforehand.
Real-World Application: This is crucial for simulating things like fuel injection in engines, the formation of clouds, or how liquids behave in micro-chips, where surface tension and chaos play a huge role.

Summary

Think of the authors as architects who built a new type of bridge (the numerical scheme) to cross a river of chaos (the difficult fluid equations).

They designed the bridge to follow strict physics laws (conservation of mass and energy).
They proved that no matter how much the wind blows (mathematical chaos), the bridge won't collapse.
They showed that as they make the bridge planks smaller and smaller, the bridge becomes indistinguishable from the "ideal" path across the river.

They didn't just build the bridge; they wrote the engineering manual proving it's safe to cross, even in a storm.

1. Problem Statement

The paper addresses the numerical approximation and convergence analysis of the Euler–Korteweg (EK) and Navier–Stokes–Korteweg (NSK) systems. These equations model compressible fluids with capillarity effects (diffuse interfaces), which are crucial for describing phenomena like phase transitions and surface tension.

The system is given by:
$\begin{aligned} \partial_t \rho + \text{div}(\rho u) &= 0, \\ \partial_t (\rho u) + \text{div}(\rho u \otimes u + p(\rho)I) &= \text{div} S(u) + \kappa \rho \nabla \Delta \rho, \end{aligned}$
where $\rho$ is density, $u$ is velocity, $p(\rho)$ is pressure, $S(u)$ is the viscous stress tensor (present in NSK, absent in EK), and $\kappa > 0$ is the capillarity coefficient.

Key Challenges:

Mathematical Complexity: The EK/NSK systems are hyperbolic (EK) or hyperbolic-parabolic (NSK) conservation laws. Classical solutions may not exist globally due to shock formation.
Non-uniqueness: Weak solutions to these systems are known to be non-unique, challenging the predictive power of standard weak formulations.
Numerical Analysis: Proving the convergence of numerical schemes for such systems typically requires strong solutions (which may not exist) or complex measure-valued frameworks (Young measures).

2. Methodology

A. The Numerical Scheme

The authors analyze a structure-preserving semi-discrete finite-volume scheme (previously introduced in [25]) on a 2D periodic domain ( $T^2$ ).

Discretization: The domain is divided into a Cartesian grid. The solution is approximated by piecewise constant functions $(\rho_h, u_h)$ .
Fluxes: The scheme uses central differences for convective terms and specific discrete approximations for the third-order Korteweg stress term ( $\kappa \rho \nabla \Delta \rho$ ).
Stabilization: A numerical dissipation term involving a parameter $\lambda$ is added to the continuity and momentum equations to ensure stability.
Conservation: The scheme is designed to conserve total mass and total momentum exactly.
Energy Dissipation: With a specific choice of $\lambda$ (related to the local sound speed and velocity), the scheme satisfies a discrete energy-dissipation inequality.

B. The Solution Concept: Energy-Variational Solutions

Instead of relying on classical weak solutions or measure-valued solutions (which involve defect measures), the authors utilize energy-variational solutions.

Definition: A triple $(\rho, m, E)$ where $E(t)$ is an auxiliary variable representing an upper bound on the total energy.
Formulation: The solution satisfies a variational inequality that combines the weak form of the equations with the energy-dissipation inequality.
$\left[ E - \int (\rho \psi + m \cdot \phi) \right] + \int (\text{weak terms}) + \int K(\phi)(E(\rho, m) - E) \leq 0$
Advantages:
- It avoids the introduction of Young measures.
- It relies on a weak-strong uniqueness principle: if a strong solution exists, the energy-variational solution coincides with it.
- The formulation is stable under weak convergence in natural energy spaces.

C. Convergence Analysis Strategy

The proof of convergence follows these steps:

A Priori Estimates: Derive uniform bounds for the discrete solutions $(\rho_h, u_h, E_h)$ based on the discrete energy inequality. This includes bounds in $L^\infty(L^\gamma)$ for density, $L^\infty(L^2)$ for momentum, and $L^2(H^1)$ for velocity (in the viscous case).
Compactness: Use the Aubin-Lions lemma and discrete Rellich-Kondrachov theorems to extract a convergent subsequence.
- Strong convergence is obtained for density $\rho_h$ .
- Weak convergence is obtained for momentum and gradients.
Limit Passage: Pass to the limit $h \to 0$ $h \to 0$ in the discrete formulation.
- Linear terms converge via weak convergence.
- Nonlinear terms (convection, pressure, Korteweg stress) are handled using the convexity of the energy functional and the specific structure of the energy-variational inequality.
- The "defect" terms arising from the nonlinearity are absorbed into the auxiliary energy variable $E$ and the regularity weight $K$ .

3. Key Contributions

First Convergence Result without Strong Solutions: This is the first paper to prove the convergence of a numerical scheme for the EK and NSK equations without assuming the existence of a strong solution.
Application of Energy-Variational Solutions: The authors successfully extend the concept of energy-variational solutions (previously used for Euler and viscoelastic fluids) to the capillary fluid systems (EK/NSK).
Structure-Preserving Scheme Analysis: They provide a rigorous convergence proof for a specific finite-volume scheme that preserves mass, momentum, and satisfies a discrete energy inequality.
Avoidance of Measure-Valued Solutions: The analysis demonstrates that energy-variational solutions are sufficient to identify the limit of numerical schemes, avoiding the complexity of Young measures or dissipative measure-valued solutions.

4. Main Results

Theorem 7 (Convergence):
Under the assumption that the discrete solutions exist on a uniform time interval and the mesh size $h \to 0$ :

There exists a subsequence of the numerical solutions $(\rho_h, u_h, E_h)$ that converges to a limit $(\rho, m, E)$ .
The limit $(\rho, m, E)$ is an energy-variational solution to the EK (if $\mu=\eta=0$ ) or NSK (if $\mu, \eta > 0$ ) system.
Convergence Rates/Types:
- $\rho_h \to \rho$ strongly in $C([0, T]; L^p)$ .
- $\rho_h u_h \rightharpoonup m$ weakly in $L^\infty(L^{2\gamma/(\gamma+1)})$ .
- If viscosity is present ( $\mu, \eta > 0$ ), $u_h \rightharpoonup u$ weakly in $L^2(H^1)$ , and $m = \rho u$ .
- The energy defect is controlled by the auxiliary variable $E$ .

Regularity Weight:
The proof relies on a specific regularity weight $K(\phi)$ involving the negative semi-definite part of the symmetric gradient and the divergence, which ensures the lower semicontinuity of the variational inequality.

5. Significance

Theoretical Validation: The work provides a rigorous mathematical foundation for using structure-preserving finite-volume schemes in simulating complex capillary fluid flows, ensuring that the numerical results converge to a physically meaningful solution concept.
Robustness: By utilizing energy-variational solutions, the framework is robust against the non-uniqueness issues inherent in weak solutions of hyperbolic systems.
Numerical Reliability: The results justify the use of these specific schemes in practical simulations where strong solutions are not guaranteed, bridging the gap between numerical practice and analytical theory.
Future Directions: The paper opens the door for applying energy-variational solution concepts to other complex fluid models and higher-order numerical schemes.

In summary, the paper establishes that a specific, physically motivated finite-volume scheme converges to a generalized solution (energy-variational) for compressible capillary fluids, providing a new, robust framework for the numerical analysis of these challenging PDE systems.