Entropy Conservative Fluctuations

• Entropy conservative fluctuations are techniques that construct interface fluxes ensuring the discrete conservation of entropy in nonlinear hyperbolic systems.
• They employ methods such as Tadmor-type path integrals and two-point flux decompositions to achieve high-order accuracy and system-independence.
• They enable precise entropy control by blending conservative formulations with dissipation mechanisms for shock capturing and well-balanced equilibria.

Entropy conservative fluctuations are a central tool in the construction of high-order numerical methods for nonlinear hyperbolic systems that respect a discrete entropy conservation law. These fluctuations are designed to ensure that the semi-discrete formulation of a system preserves the underlying thermodynamic entropy in smooth regions, while allowing controlled entropy dissipation at discontinuities through entropy-stable extensions. The concept is foundational in the development of entropy stable discontinuous Galerkin (DG), finite difference, and finite volume methods, with significant advances enabling system-independent algebraic constructions and high-order accuracy for both conservative and nonconservative systems (Ersing et al., 14 Jan 2026, Chan, 2017).

1. Entropy Structure and Conservation Laws

A nonlinear hyperbolic balance law is generally written as

$u_t + f(u)_x + B(u)u_x = 0$

where $u \in \mathbb{R}^n$ is the state vector, $f(u)$ the conservative flux, and $B(u)$ a nonconservative coefficient matrix. A strictly convex entropy $S(u)$ and entropy flux $q(u)$ form an entropy pair if they satisfy the compatibility relation

$$q_u(u) = w(u)^T A(u), \quad w(u) = \partial_u S(u), \quad A(u) = f_u(u) + B(u)$$

This ensures that, for smooth solutions, the entropy conservation law

$S(u)_t + q(u)_x = 0$

holds, whereas physically admissible weak solutions must satisfy the entropy inequality

$S(u)_t + q(u)_x \leq 0$

The goal of entropy conservative fluctuations is to design numerical fluxes and interface terms such that the semi-discrete method precisely reproduces this balance at the discrete level (Ersing et al., 14 Jan 2026, Hicken et al., 2018).

2. Construction of Entropy Conservative Fluctuations

An entropy conservative fluctuation consists of a pair of interface terms $$D^-_{EC}, D^+_{EC} : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$$ that, for each interface between $u_L$, $u_R$, satisfy:

• Two-state consistency: $D^\pm(u, u) = 0$
• Path conservation: $D^-_{EC}(u_L,u_R) + D^+_{EC}(u_L,u_R)$ equals the integral of the flux Jacobian along any path connecting $u_L$ and $u_R$
• High-order volume compatibility: on differentiation, recovers the continuous $A(u)u_x$
• Skew-symmetry: $D^-_{EC}(u_L,u_R) + D^+_{EC}(u_R,u_L) = 0$
• Discrete entropy conservation:

$$w(u_L)^T D^-_{EC}(u_L, u_R) + w(u_R)^T D^+_{EC}(u_L, u_R) = q(u_R) - q(u_L)$$

General system-independent strategies to construct $D^\pm_{EC}$ include (Ersing et al., 14 Jan 2026):

1. Tadmor-type in entropy variables:

Integrate along a straight path in entropy variables,

$\Phi(s) = w_L + s(w_R - w_L)$

and define

$$D^-_{EC}(u_L, u_R) = \int_0^1 (1-s)A(u(\Phi(s))) H(u(\Phi(s))) (w_R - w_L) ds$$

$$D^+_{EC}(u_L, u_R) = \int_0^1 sA(u(\Phi(s))) H(u(\Phi(s))) (w_R - w_L) ds$$

where $H(u) = S_{uu}(u)$ is the entropy Hessian.

2. Two-point form:

Decompose $A(u)$ and use a symmetric two-point flux $f^*$ for the conservative part, along with a suitable split of the nonconservative matrix, to form

$$D^\pm_{EC}(u_L, u_R) = \frac{1}{2}\bar{B}^{\pm}(u_L, u_R) \Delta v + \frac{1}{2}(f(u_{L/R}) - f^*(u_L, u_R))$$

where $f^*$ and $\bar{B}^\pm$ are selected to satisfy the entropy conservation condition.

For conservative systems, these approaches reduce to the classic entropy-conservative two-point flux construction of Tadmor (Hicken et al., 2018, Chan, 2017).

3. Fluctuation-Splitting and the Discrete Entropy Balance

The central algebraic property of entropy conservative fluctuations is the discrete entropy conservation law. In finite volume or DG settings, using two-point entropy-conservative fluxes $f^*(u_L, u_R)$, one defines the cell interface fluctuation as

$$\Delta f_{i+1/2} = f^*(u_i, u_{i+1}) - f^*(u_{i-1}, u_i)$$

and updates the solution as

$$\frac{d u_i}{dt} + \frac{1}{\Delta x}(f^*(u_i, u_{i+1}) - f^*(u_{i-1}, u_i)) = 0$$

Multiplying by $w(u_i)^T$ and summing over $i$ yields

$$\frac{d}{dt} \sum_i S(u_i) + \sum \limits_\text{faces} [\text{entropy flux terms}] = 0$$

due to the cancellation arising from the entropy conservative constraint (Hicken et al., 2018, Gouasmi et al., 2018).

A general form of the entropy production term in the semi-discrete balance is

$$E_i = (v_{i+1}-v_i)^T\,\mathcal F^*_{i+1/2} - (\Psi_{i+1}-\Psi_i)$$

which is identically zero for EC fluxes (Michele et al., 2023).

4. Achieving Entropy Stability: Dissipation and Blending

While entropy conservative fluctuations yield zero entropy production for smooth flows, physically relevant solutions require discrete entropy dissipation at shocks. Entropy stability is enforced by augmenting the interface fluctuation with a matrix-valued numerical viscosity,

$$D^\pm_{ES}(u_L, u_R) = D^\pm_{EC}(u_L, u_R) \pm D(u_L, u_R)\frac{1}{2}\Delta u$$

where, for entropy stability,

$w^T D(u_L, u_R) \Delta u \geq 0$

A universally robust choice is the local Lax–Friedrichs (LLF) dissipative viscosity,

$D_{llf}(u_L, u_R) = \frac{1}{2}|\lambda|_{\max} I$

where $|\lambda|_{\max}$ is the maximal absolute eigenvalue of $A(u)$ along the path. For higher fidelity or well-balance properties, one can blend LLF with Roe- or HLL-type dissipative matrices:

$D_{blend} = \alpha D_{llf} + (1-\alpha) D_{other}$

with $\alpha$ set to ensure nonnegativity of entropy production; for example,

$$\alpha = \min\{1,\; \Delta S_{other}/|\Delta S_{llf}-\Delta S_{other}|\}, \quad \Delta S_{other}>0, \, 0 \text{~otherwise}$$

This guarantees entropy stability while allowing model-specific well-balanced dissipation (Ersing et al., 14 Jan 2026).

5. High-Order Discretizations and Implementation

In high-order DG or SBP formulations, entropy conservative fluctuation-based schemes rely on flux differencing within the volume and appropriate quadrature rules. For DG implementations on a reference interval with Lagrange bases at GLL points, the discrete derivative and weight matrices are constructed to satisfy the summation-by-parts (SBP) property, ensuring telescoping cancellations of internal entropy flux and exact semi-discrete entropy conservation (Chan, 2017). The semi-discrete update incorporates the entropy conservative volume flux, interface fluctuations, and, if desired, the dissipative entropy-stable extensions at element boundaries.

The implementation sequence, as outlined in (Chan, 2017), involves projection between conservative and entropy variables, assembly of the two-point flux matrices, and application of the DG operator together with entropy-stable interface corrections.

6. Numerical Evidence and Practical Observations

Numerical experiments on both conservative and nonconservative systems consistently demonstrate that entropy conservative fluctuations:

• Deliver machine-precision discrete entropy conservation in smooth regimes, provided quadrature errors are negligible.
• Maintain the designed spatial order of convergence in all primary and auxiliary flow quantities.
• Control spurious entropy fluctuations in the presence of strong gradients when augmented with dissipative corrections.
• Enable well-balanced steady-state preservation for physically relevant equilibria when paired with appropriately blended dissipation matrices (Ersing et al., 14 Jan 2026, Michele et al., 2023).

For example, in DG simulations of the Saint-Venant–Exner system, both path-integral and closed-form two-point EC fluctuations yield vanishing discrete entropy residuals in smooth tests, while entropy-stable blending with Roe-type viscosity preserves both entropy decay and well-balanced equilibria in the presence of shocks and discontinuous steady states.

7. System-Independence and Extensions

A fundamental advancement of recent work is the elimination of model-specific symmetrization or ad hoc modifications in the construction of entropy conservative fluctuations (Ersing et al., 14 Jan 2026). Both the entropy-variable path integral (Tadmor-type) and algebraic two-point ansatz provide universally applicable frameworks for any system admitting a convex entropy pair. These constructions extend to thermally perfect gases, multicomponent Euler systems, and nonconservative hyperbolic systems, as illustrated by (Aiello et al., 10 Jul 2025) and general recipes outlined in (Michele et al., 2023). Asymptotically entropy conservative (AEC) extensions further enable polynomial or algebraic approximations of the logarithmic mean-based EC fluxes, with controlled reduction of entropy errors in the mesh refinement limit (Michele et al., 2023).

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Published results confirm that entropy conservative fluctuations are a robust, extensible backbone for high-fidelity, stable simulation of nonlinear hyperbolic systems, underpinned by rigorous algebraic entropy analysis and supported by universal construction principles (Ersing et al., 14 Jan 2026, Chan, 2017, Hicken et al., 2018, Michele et al., 2023, Aiello et al., 10 Jul 2025, Gouasmi et al., 2018).