# A 1D-2D Coupled Lattice Boltzmann Model for Shallow Water Flows in Large Scale River-Lake Systems
## 1 Introduction  
One-dimensional (1D) and two-dimensional (2D) shallow water equations are widely used to simulate flows in rivers, lakes, reservoirs, and estuaries. The dimensionally different models are suitable for different scenarios. The cross-sectional integrated 1D models are extremely suitable for hydrodynamic simulations in large scale river networks, with their lumped boundary representations for hydraulic structures, such as weirs, dams, culverts, and pumps. On the other hand, 2D models, with their ability to accurately resolve flows in both longitudinal and transversal directions, are the primary choice for hydrodynamic simulations for lakes, reservoirs, and estuaries. Because 1D river networks and 2D lakes are often linked together, it is natural to choose the 1D-2D coupled model for accurately investigating their interactions. The numerical solvers for the shallow water equations are mainly based on traditional methods, such as an implicit finite-difference scheme for the 1D model and finite-volume method for the 2D model [1; 2; 3]. Even if being accepted widely, these traditional methods are relatively difficult toimplement, and the calculations are usually time-consuming. Recently, the lattice Boltzmann (LB) method is developed for simulating complicated fluid flow problems, such as multi-dimensional hydrodynamic flows [4; 5], multiphase flows [6; 7], porous media flows, and fluid-structure interaction problems [8]. Its easy implementation, intrinsic parallelism, and high accuracy have been demonstrated by many works [5; 8; 9]. Because of these advantages, introducing the LB method into simulations of the shallow water flows seems to be very attractive. [PERSON] [10; 11] first introduced a 1D LB model for the 1D shallow-water equations, and [PERSON] [12] had further verified the model. Later, [PERSON] [13] used it to simulate flows in complex irrigation channels, and [PERSON] [14] used it to simulate flows in rapids canals. On the other hand, [PERSON] [15] first developed a 2D LB model for the shallow water equations, in which the terrain and riverbed friction were represented by source terms in the original LB equation. [PERSON] and [PERSON] [4; 9; 16] used a similar 2D LB model to simulate the surge propagations caused by the transient processes of hydropower stations. [PERSON] [17; 18; 19] adopted the 2D models to simulate flows in open channels and shallow lakes. This paper aimed to propose a 1D-2D coupled model for simulating shallow water flows in river-lake systems. Thus, it is necessary to discuss the 1D-2D coupling strategies. According to the specific problems, the strategies are mainly divided into three categories: the lateral coupling, the superposition coupling, and the boundary connected coupling [20]. The first category is mainly adopted to simulate flows in floodplain systems [2; 21; 22]. The second category is mainly used to simulate flows in systems with obvious 2D flow effects of local flow patterns, in which the high-dimensional model is superimposed on the low-dimensional global model [23; 24]. The third category is mainly applied to rivers, lakes, reservoirs, and estuarine systems, which connect different dimensional models directly through the coupling boundaries. There are a variety of coupling strategies available for the third category. The simplest one is to run each submodel in sequence, with its boundary conditions provided by the previous solution of each other [25; 26]. However, the mutually provided boundary conditions are lagged, causing the non-conservation of mass or momentum. To reduce the errors, overlapped areas are set at the coupling boundaries [27; 28; 29], but it still cannot avoid the errors caused by the lagged boundary conditions. Another strategy is to use iterative methods to solve the coupling boundaries, which can fully meet the conservation of mass and momentum [20; 30].