# An AMR-Based Liquid Film Simulation with Surfactant Transport Using PLIC-HF Method
## 1 Introduction  
Following the development of the numerical fluid simulation field, a wide range of fluid phenomena can now be simulated. Regarding multiphase flows, the generation and rupture of thin liquid film has attracted researchers' interest. Thin liquid films are ubiquitous in many naturally occurring processes as well as in many industrial processes [1]. Typically, when a bubble starts rising from liquid and approaches the interface, a thin liquid film is generated by the drainage of liquid. The dynamics of this process determine the behaviors of fluid in many liquid-gas two-phase flows. Due to its practical importance, the dynamics and stability of thin liquid film have been widely researched through experiments and theoretical analyses [2; 3; 4; 5; 6; 7]. However, there has been little research on the dynamics and stability of thin liquid film using numerical simulations. The difficulty of conducting liquid film simulations can be divided into several aspects. First, in a thin liquid film simulation, the length of the problem is typically much larger than the thickness of the film, which makes it a multiscale problem. In simulations of such multiscale problems, a very-high-resolution mesh grid is required to ensure that an accurate description of the behavior of liquid film is achieved. With a uniform resolution mesh grid and CPU computing, the required memory of the computer and execution time make simulations of thin liquid film impractical. CPU programs are used in conventional numerical simulations. However, the CPU is a single processing unit that can only executea limited number of instructions at a time. Thus, in large-scale simulations, or simulations of thin liquid film that require a high-resolution mesh grid, the efficiency of the CPU is unsatisfactory. To accelerate the computation process, GPU parallel computing can be adopted. The GPU is a specialized type of processor that was designed to handle a large number of parallel calculations simultaneously. This makes GPUs well-suited for certain types of computational tasks, such as the rendering of graphics or the performance of complex calculations for scientific simulations. The GPU can provide over 100 times more FLOPS than the CPU in large-scale simulations [8]. In the simulation of real phenomena, it is inefficient to use a uniform resolution grid in the computational domain. Normally, the resolution of the grid is determined by the size of the cells or elements that make up the grid. The size of these cells is typically chosen based on the size of the smallest features in the system being simulated as well as the desired level of simulation accuracy. For example, if a simulation is being performed to study the flow of a fluid through a complex network of pipes, the grid resolution needs to be fine enough to accurately capture the details of the pipe geometry and the flow of the fluid within the pipes. This requires the use of small grid cells in the regions of the grid that correspond to the pipes to accurately represent their shapes and sizes. However, in other regions of the grid, where the fluid flow is less complex, the grid cells can be larger, as the details of the flow are not as important. This is a basic concept of the Adaptive Mesh Refinement (AMR) method. In the 1980s, led by [PERSON] [9], researchers proposed the AMR method to dynamically adjust the size of cells and the resolution of the grid based on the complexity of the solution with higher resolutions used in regions with rapidly changing solutions and lower resolutions in regions with relatively constant solutions. In previous work by [PERSON] and [PERSON], the execution time and size of the computational grids were reduced by around 90% through the use of a benchmark advection problem from the interface-capturing method [10; 11]. Second, a highly accurate interface capturing method for two-phase flows is required for the simulation of thin liquid films.