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

TL;DR

This paper develops a second-order generalized Riemann problem solver for a two-layer thin film model, improving accuracy and efficiency over existing methods through explicit derivative computation and robust numerical experiments.

Contribution

The paper introduces a novel second-order GRP solver tailored for a two-layer thin film model, enabling explicit and computationally efficient derivative calculations.

Findings

The second-order GRP solver outperforms first-order methods in accuracy.

Numerical experiments confirm the robustness and efficiency of the proposed method.

Explicit derivative computation simplifies the implementation for the two-layer thin film system.

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.