sustainability Article A Hydrodynamic-Based Robust Numerical Model for Debris Hazard and Risk Assessment Yongde Kang, Jingming Hou *, Yu Tong and Baoshan Shi State Key Laboratory of Eco-Hydraulics in Northwest Arid Region of China, School of Water Resources and Hydroelectric Engineering, Xi’an University of Technology, Xi’an 710048, China; [email protected] (Y.K.); [email protected] (Y.T.); [email protected] (B.S.) * Correspondence: [email protected] Abstract: Debris flow simulations are important in practical engineering. In this study, a graphics processing unit (GPU)-based numerical model that couples hydrodynamic and morphological pro- cesses was developed to simulate debris flow, transport, and morphological changes. To accurately predict the debris flow sediment transport and sediment scouring processes, a GPU-based parallel computing technique was used to accelerate the calculation. This model was created in the frame- work of a Godunov-type finite volume scheme and discretized into algebraic equations by the finite volume method. The mass and momentum fluxes were computed using the Harten, Lax, and van Leer Contact (HLLC) approximate Riemann solver, and the friction source terms were calculated using the proposed splitting point-implicit method. These values were evaluated using a novel 2D edge-based MUSCL scheme. The code was programmed using C++ and CUDA, which can run on GPUs to substantially accelerate the computation. After verification, the model was applied to the simulation of the debris flow process of an idealized example. The results of the new scheme better reflect the characteristics of the discontinuity of its movement and the actual law of the evolution Citation: Kang, Y.; Hou, J.; Tong, Y.; of erosion and deposition over time. The research results provide guidance and a reference for the Shi, B. A Hydrodynamic-Based in-depth study of debris flow processes and disaster prevention and mitigation. Robust Numerical Model for Debris Hazard and Risk Assessment. Keywords: debris flow; Godunov-type scheme; numerical model; graphics processing unit (GPU) ac- Sustainability 2021, 13, 7955. https:// celeration doi.org/10.3390/su13147955 Academic Editors: Ashraf Dewan and Jui-Sheng (Rayson) Chou 1. Introduction Debris flows have occurred in many provinces and municipalities in China, and they Received: 29 May 2021 can be disastrous, often destroying nearly everything in their path and threatening lives, Accepted: 8 July 2021 Published: 16 July 2021 property, and infrastructure [1–3]. It is generally believed that a debris flow is a series of processes between the movement of blocks such as collapse and landslide and sediment- Publisher’s Note: MDPI stays neutral bearing water flow, which is formed by the interaction and development of solid-liquid with regard to jurisdictional claims in two-phase materials on mountain slopes or channels. published maps and institutional affil- At present, most debris flow models can be clarified as dynamic models and numeri- iations. cal models. The dynamic model of debris flow can be divided into continuous medium, discrete medium, and mixed medium models from the perspective of describing the com- position and movement. Continuous medium models can be divided into single-fluid [4,5] and multifluid models [6–10]. The two-phase model is attractive because it can explicitly reveal the relative motions and interactions between the fluid and solid phases. Accord- Copyright: © 2021 by the authors. ingly, however, the increase in computing costs and also the demand for extra relationships Licensee MDPI, Basel, Switzerland. This article is an open access article to close the governing equations constrain its applications. Moreover, it is still unclear distributed under the terms and if they can perform considerably better than traditional single-phase models in terms of conditions of the Creative Commons modelling accuracy [11]. Single fluid models are generally applicable to debris flows with a Attribution (CC BY) license (https:// large bulk density or a small velocity difference between two phases [12]. Non-Newtonian creativecommons.org/licenses/by/ models include the Bingham body [13], the Bagnold expansion body [14], and the viscoelas- 4.0/). tic body [15]. The multifluid model mainly refers to the two-component hydrodynamic Sustainability 2021, 13, 7955. https://doi.org/10.3390/su13147955 https://www.mdpi.com/journal/sustainability Sustainability 2021, 13, 7955 2 of 19 equations established by considering the momentum exchange between solid and liquid phases. These models can explicitly reveal the relative motion and interaction between the fluid and the solid phase [16]. For example, in the Coulomb mixed flow model, considering the constitutive relationship of granular matter and the interaction force between two phases [17–19], Pitman and Le [20] proposed a two-fluid model considering the effect of liquid buoyancy. The discrete medium model simplifies the debris flow into a sys- tem composed of a large number of material particles of a certain size. In recent years, with the development of thermal dynamics theory and turbulence theory, the traditional Boltzmann equation has been extended to the study of turbulence and particle flow [21], which provides a method different from the traditional Navier–Stokes (N-S) equation for the theoretical and numerical study of debris flows. The mixed medium model uses a continuous medium model and a discrete medium model to describe the movement of the liquid and solid parts of debris flows. In practice, debris flows are composed of two phases; only certain special types of debris flows, such as mud flows, can be simplified as a phase flow. Although the multifluid model describes the liquid phase and the solid phase with different equations, the equation describing the motion of solid particles is still a continuum model. Generally, the liquid phase of debris flows is a slurry liquid composed of fine particles and water that is smaller than a certain size, which can be regarded as a continuous medium. The solid phase of large particles above a certain size should be described by a discrete medium model. The liquid phase of debris flows is described by a non-Newtonian continuous medium, and the motion equation adopts the two-dimensional depth-averaged shallow water wave equation. The numerical calculation model of debris flow is closely related to the research problems. Due to the complexity of the debris flow process, a number of models were de- veloped to simulate the flow behavior. The first numerical calculations of debris flows were performed using a one-dimensional homogeneous single-phase continuum model [22–24]. With the development of computational fluid dynamics, numerical calculation methods, computer technology, and debris flow theory, numerical approaches have been gradually extended to include multidimensional, heterogeneous, and multiphase models [25–33]. It is well known that numerical simulations are an important tool for studying de- bris flow disaster process. Although full three-dimensional models [34] can increase the precision of high-resolution simulations and may help very detailed resolution of the phenomena, when calculating large practical cases, this method is impractical due to the high calculation cost. In contrast, using the principles of the conservation of mass and momentum, depth-averaged models provide a reasonable balance of completeness and theoretical application and have therefore been broadly applied. Future science and engineering breakthroughs depend on computing. In recent years, through the implementation of parallelization techniques, there has been a reliable way to significantly reduce computational effort, such as multiprocessing (open MP) and message passing interface (MPI), which allow simulations to be run on cluster machines [35]. Graph- ics processing units (GPUs) are a new computation engine for high-resolution modeling as computation burdens become increasingly heavy. To resolve the challenge of further accelerating up computations, modern high-performance computer systems increasingly employ GPUs and other accelerators, such as open MP and MPI. Their disadvantages are related hardware cost and energy processor requirements, which usually create a limitation on their practical usage. On the contrary, hardware accelerators, such as GPU, have become a low-cost option, because they can be used on simple personal computers. Previous studies have developed strategies for implementing the pure shallow water equations on graphics processors [36,37]. However, less research has focused on high-resolution terrain using GPU acceleration technology. GPU computing is expected to become more common in future systems. Therefore, programs must be adapted to the use of graphics processors, because graphics processors provide high performance with low costs and power require- ments, and therefore they have become the main computing resources of many of the largest supercomputers. Under the framework of a shallow sediment-geomorphology dynamic Sustainability 2021, 13, 7955 3 of 19 model, this study focused on a two-dimensional depth-averaged quasi-multiphase mixing model with non-uniform sediment transport to simulate the evolution of debris flows on inclined bed slopes. Namely, by using the GPU Accelerated Surface Water Flow and Transport Model (GAST) model, which can accurately predict debris sediment transport and the debris sediment scouring process, GPU techniques were applied in a numerical model, making it possible to simulate the