A generalized Riemann problem solver for a hyperbolic model of two-layer thin film flow

TL;DR

This work develops a second-order generalized Riemann problem (GRP) solver for a four-variable hyperbolic system modeling two-layer thin film flow, written in conservative form as $\mathbf{U}_t+(\mathbf{F}(\mathbf{U}))_x=0$ with $\mathbf{U}=(f,b,g,q)^T$. By exploiting a decoupling along Riemann invariants, the authors derive explicit time derivatives of the conservative variables via Rankine–Hugoniot relations and a diagonalized (or near-diagonal) representation, enabling a temporally and spatially coupled GRP-based finite-volume scheme. The method is implemented for two principal wave configurations, providing linear systems to compute instantaneous derivatives across rarefaction, contact, and shock waves, with explicit acoustic/sonic subcases. Numerical tests show that the GRP scheme achieves higher accuracy and efficiency than MUSCL–RK2 and Godunov methods in both smooth travelling waves and discontinuous scenarios, and suggest potential extensions to entropy-stable discontinuous Galerkin methods and higher dimensions. This approach offers a practical, high-order tool for simulating hyperbolic thin-film dynamics with explicit derivative information embedded in the Riemann-invariant structure.

Abstract

In this paper, a second-order generalized Riemann problem (GRP) solver is developed for a two-layer thin film model. Extending the first-order Godunov approach, the solver is used to construct a temporal-spatial coupled second-order GRP-based finite-volume method. Numerical experiments including comparisons to MUSCL finite-volume schemes with Runge-Kutta time stepping confirm the accuracy, efficiency and robustness of the higher-order ansatz. The construction of GRP methods requires to compute temporal derivatives of intermediate states in the entropy solution of the generalized Riemann problem. These derivatives are obtained from the Rankine-Hugoniot conditions as well as a characteristic decomposition using Riemann invariants. Notably, the latter can be computed explicitly for the two-layer thin film model, which renders this system to be very suitable for the GRP approach. Moreover, it becomes possible to determine the derivatives in an explicit, computationally cheap way.

Figures (9)

• Figure 1: Flowchart of the Riemann solver for the system \ref{['eq: Main_system']} that leads to four different wave patterns in Case 1(a), 1(b), 2(a) and 2(b). Here $S_i, R_i$ for $i\in \{1, 4\}$ denotes the $i$-shock and $i$-rarefaction wave, respectively. The symbol $J_i$ for $i\in \{2, 3\}$ stands for an $i$-contact wave.
• Figure 2: Comparison of the GRP method with the Godunov method for Example \ref{['pure_rarefaction']} at $t=2.5$ with $\rm{CFL}=0.4$ and $N=100 ~(\Delta x=0.25)$.
• Figure 3: Comparison of the GRP method and the MUSCL-RK2 scheme for Example \ref{['example:7.3']}
• Figure 4: Comparison of the GRP method with the MUSCL-RK2 scheme for Example \ref{['large_ratio']} with $N=100 (\Delta x=0.25)$ and $\rm{CFL}= 0.4$ at time $t=5.0$.
• Figure 5: Comparison of the GRP method with the second-order MUSCL scheme with RK-2 time stepping for the shock-tube problem at $t=2.5$ with $N=100 (\Delta x=0.25)$ and $\rm{CFL}=0.4$.

Theorems & Definitions (30)

• Proposition 3.1
• proof
• Remark 3.1
• Lemma 3.1
• proof
• Remark 3.2
• Remark 3.3
• Remark 3.4
• Lemma 3.2
• Lemma 3.3