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Entropy dissipative high order schemes for a hyperbolic model of two-layer thin film flow

Rahul Barthwal, Philipp Öffner, Chetan Singh, Jan Zawallich

arXiv:2610.02059v1math.NAcs.NAmath.AP

TL;DR

Entropy structure helps identify physically admissible solutions of the hyperbolic thin-film system, motivating entropy-preserving discretizations. The paper constructs exact and Taylor-approximated entropy-conservative fluxes for FD and DGSEM schemes; the exact formulation is more efficient in the numerical schemes, while the flux choice affects solver efficiency.

  • Problem

    Entropy structure is needed to select physically admissible weak solutions when the hyperbolic thin-film subsystem can develop shock-like discontinuities, and extending entropy-stable methods to coupled multi-layer systems remains open.

    Source: §1 · pp. 1–2

  • Method

    The paper derives exact and Taylor-approximated entropy-conservative fluxes and incorporates them into entropy-dissipative finite-difference and DGSEM frameworks.

    Source: §1 · p. 2, Entropy Dissipative Discontinuous… · p. 6, §3.2 · p. 9

  • Results

    The analytically derived flux can be evaluated exactly within each time step and yields a more efficient approach in numerical schemes than the Taylor-approximated flux, which must be solved numerically.

    Source: §5 · p. 22

  • Takeaways & Limitations

    Flux construction affects computational cost, robustness, and stability of the resulting solvers, in both FD and DGSEM discretizations.

    Source: §1 · p. 2

    The analytical-flux system is uniquely solvable only when det(Ã)≠0.

    Source: §3.3 · p. 11

Abstract

from arXiv · show

In this article, we develop high-order, entropy-dissipative schemes for a hyperbolic system of conservation laws describing the first-order dynamics of two-layer thin-film flows of immiscible fluids with perfectly soluble solute particles. The key aspect is to construct entropy-conservative fluxes in the sense of Tadmor. Here, we consider and compare two different construction processes: an exact formulation and a commonly used simplified approximation in the flux evaluation. While the approximate approach is simpler at the level of construction and is frequently employed in practice (also in other contexts), it is less structured, leading to increased computational cost in the implementation compared to the exact formulation, which has been much more complicated to derive. We employ these fluxes within both an entropy-dissipative finite-difference framework and an entropy-dissipative discontinuous Galerkin spectral element method. Through numerical experiments using two distinct spatial discretizations, we investigate the performance and robustness of the proposed methods and demonstrate that the choice of numerical flux has a noticeable impact on the efficiency of the resulting solvers.

1 Introduction

Entropy provides a criterion for selecting physically admissible solutions when the two-layer thin-film system develops shocks, but extending entropy-stable methods to coupled multi-layer systems remains open. This work constructs two entropy-conservative fluxes and compares their use in finite-difference and discontinuous Galerkin schemes.

1 Introduction

  • Entropy inequalities select physically admissible weak solutions, which matters because shock-like structures can arise from smooth initial data.The entropy structure is therefore a key ingredient in stable, physically consistent discretizations.Source: §1 · pp. 1–2
  • Entropy-stable methods for thin films have been developed for scalar single-layer equations, while extension to coupled single- and multi-layer systems remains largely open.Prior developments span finite-difference and finite-element-type frameworks and build on Tadmor’s entropy-conservative flux framework.Source: §1 · p. 2
  • The paper derives approximate and exact entropy-conservative two-point fluxes and uses them in entropy-dissipative FD and DGSEM schemes for two-layer thin-film flow.The exact flux preserves additional symmetry in evaluation, while the approximate construction is simpler; the authors compare solver efficiency and robustness across both discretizations.Source: §1 · p. 2

2 Model

The paper formulates the thin-film model as a first-order hyperbolic system and establishes its structural properties, including a generalized state space and a sign-independent strictly convex entropy.

2 Model

  • The conservative state U=(f,b,g,q)ᵀ uses positive layer thicknesses f and g, while b and q are spatial derivatives of solute concentrations in the two layers.The system is written in conservative form with a corresponding flux vector.Source: §2 · p. 3
  • The state space requires f,g,q>0, b≠0, gq>fb, and 2fb+gq>0, generalizing earlier sign-restricted domains to allow either sign of b.For b>0 and b<0, it reduces to the respective state spaces considered in [7] and [29].Source: §2 · p. 3
  • The system’s eigenvectors are linearly independent in the state space, implying that the reduced system is strongly hyperbolic.Source: §2 · p. 3

2.1 Entropy/entropy pairs of the hyperbolic system (1)

  • The authors first obtain a strictly convex entropy for system (1) independent of the sign of b, generalizing while simplifying the convex entropy obtained in [7].Source: §2.1 · p. 4

3 Constructing high-order entropy dissipative schemes

This section builds higher-order entropy-dissipative schemes around entropy-conservative two-point fluxes, presenting Taylor-based and analytical constructions for use in DGSEM and finite differences.

3.1 Notation and background

  • The setup defines jump and average notation and introduces Tadmor’s entropy-conservative-flux condition and the entropy potential for the stated entropy pair.Source: §3.1 · p. 6

Entropy Dissipative Discontinuous Galerkin Spectral Element Method

  • In DGSEM, flux differencing uses two-point volume and surface fluxes to preserve conservation and accuracy; entropy-conservative volume fluxes yield elementwise entropy conservation, while interface flux choice sets conservation or dissipation.Source: Entropy Dissipative Discontinuous… · pp. 6–7

Entropy Dissipative Finite Difference Scheme

  • The finite-difference scheme adds dissipation to entropy-conservative fluxes, then uses sign-preserving reconstruction of entropy-scaled variables to raise the basic first-order flux to higher order while retaining entropy stability.Source: Entropy Dissipative Finite Difference… · pp. 8–9

3.2 Entropy conservative flux using Taylor expansion

  • When a_l≈a_r, the Taylor approach approximates the logarithmic mean, expresses entropy-variable jumps through variable jumps, and solves a 4×4 linear system numerically at each time step.Source: §3.2 · pp. 9–10

3.3 Analytical derivation of numerical flux

  • The analytical approach rewrites Tadmor’s condition in conservative variables and derives a linear system for the flux; when det(Ã)≠0, its solution gives an analytical entropy-conservative two-point flux.Source: §3.3 · pp. 10–11

4 Numerics

The section evaluates entropy-dissipative FD and DGSEM schemes on a smooth wave, two one-dimensional Riemann problems, and two-dimensional cases, comparing analytical and Taylor fluxes for accuracy, robustness, and efficiency.

4.1 Smooth traveling wave solutions - convergence test

  • DGSEM errors decrease at order h^(p+1); Taylor flux has a slightly larger error constant, while analytical flux is at least one order of magnitude faster and avoids p=1 breakdown.The reported speed difference reaches two orders of magnitude; each Taylor-flux evaluation requires a 4 × 4 matrix inversion.Source: Figure 1 · p. 12, §4.1 · pp. 12–13
  • Third-order FD runtime on the finest mesh falls from 125.0 to 98.9 with analytical flux, about 21%; both fluxes achieve expected third- and fourth-order convergence.The analytical flux becomes consistently more efficient on finer grids; reported reductions are about 11% and 7% for second- and fourth-order schemes.Source: Figure 2 · p. 12, §4.1 · p. 13

4.2.1 A shock-tube type initial data

  • For the shock-tube FD test, analytical and Taylor profiles differ little beyond order; both entropy changes are negative: −6.21110298 × 10−2 (Taylor) and −6.21110521 × 10−2 (analytical).The negative values confirm entropy dissipation for this test.Source: §4.2.1 · pp. 14, 16

4. Numerics

  • 0.078 versus approximately 1.573 seconds for shock-tube DGSEM, and around 21 versus 860 seconds in 2D, favor the analytical flux with visually similar solutions.The 2D Taylor-flux runtime reflects the reported implementation, which used an in-house solver without further optimization.Source: Figure 4 · p. 16, §4.2.1 · p. 16, §4.3 · p. 20

4.2.1 A shock-tube type initial data

  • The f4 smoothness indicator robustly locates discontinuities and stabilizes subcell reconstruction here, but constructed cases can defeat it; subcell FV is dissipative, with refinement improving sharp-gradient resolution.Small oscillations remain visible in the 128-cell result.Source: §4.2.1 · pp. 16–17, Figure 5 · p. 17

4.2.2 Riemann problem with negative b

  • For the Riemann problem with negative b, the adapted fluxes are tested with FD and DGSEM; small oscillations appear in f and are smaller in the other variables.The DGSEM results use polynomial degrees p = 2 and p = 3.Source: §4.2.2 · p. 17, Figure 6 · p. 18

4.3 Two-dimensional problems

  • In 2D, the analytical-flux DGSEM reproduces the reported circle motion, while a 128 × 128 grid resolves ring-test jumps in f and g well.The ring test has b strictly negative and q opposite in sign up to a positive factor.Source: §4.3 · pp. 20, 22, Figure 8 · p. 21, Figure 9 · p. 22

5 Outlook and Summary

The paper compares Taylor-approximated and analytically derived entropy-conservative fluxes in entropy-dissipative finite-difference and DGSEM methods. Their accuracy and stability are similar, but the analytical flux reduces runtime; future work will pursue bound-preserving limiters.

5 Outlook and Summary

  • Both fluxes give similar accuracy and stability, while the analytical flux improves runtime by allowing exact evaluation instead of a numerical solve at each time step.The comparison uses entropy-dissipative finite-difference and flux-differencing DGSEM methods across smooth problems, one-dimensional Riemann problems, and two-dimensional problems.Source: §5 · p. 22
  • The Taylor-approximated flux is easier to derive, whereas the analytical flux is evaluated exactly within each time step.Both approaches lead to a linear system, but only the Taylor-approximated flux requires that system to be solved numerically.Source: §5 · p. 22
  • Shock experiments use additional limiters in the finite-difference method and a subcell finite-volume technique in DGSEM.The DGSEM subcell technique is developed in [36].Source: §5 · p. 22
  • Future work will develop bound-preserving limiters to provably ensure preservation of the model's physical laws.The planned limiters are described as being in the spirit of [40] [41].Source: §5 · p. 22

A Details for Taylor expansion

The Taylor-expansion derivation expresses entropy-variable jumps through jumps in the model variables, then solves a linear system for the entropy-conservative flux components.

A Details for Taylor expansion

  • The Taylor-based construction uses a Taylor expansion and the discrete chain rule to express the needed jumps.The derivation organizes these expansions by jumps in the conservative variables and in the potential.Source: A Details for Taylor expansion · pp. 25–26
  • Equating coefficients of f, b, g, and q in (31) and (32) yields a 4 × 4 linear system, AF_num = b, with its coefficient matrix and right-hand side given explicitly below.Source: A Details for Taylor expansion · p. 27
  • The Taylor-approximation approach requires solving its linear system numerically at each time step to calculate the entropy-conservative flux.The corresponding exact form is provided in the repository.Source: A Details for Taylor expansion · p. 28

B Detail for analytical derivation

The analytical derivation uses the entropy potential and jump relations to obtain the entropy-conservative numerical flux through a linear system.

B Detail for analytical derivation

C Entropy-scaled right eigenvectors for two-layer thin film flow model

The appendix constructs entropy-scaled right-eigenvector matrices using Barth scaling and gives separate expressions for the two signs of b.

C Entropy-scaled right eigenvectors for two-layer thin film flow model

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