## 1 Introduction  
In the tsunami generation, \"landslide\" is a general term that describes several types of mass movement, including rock falls, slope failures, debris flows, slides, and slumps [1]. An accurate numerical simulation of landslide-generated tsunamis plays an important role in understanding the generation and the propagation of water waves, and in predicting a natural disaster [2]. For the past few decades, several analytical and empirical solutions have been proposed to calculate the wave height [3, 4], the velocity of the submerged solid [5, 6], wave velocity along a plane beach [7], and the dynamics of tsunamis generated and propagated by submarine landslides [8]. However, the analytical and empirical solutions have difficulty in describing the detailed wave and flow fields due to the complex flow conditions, such as the strong turbulence generated during the sliding process, the strong wave breaking, and the generation of intermedia to short waves at the near-source region, not to say the difficulties in monitoring and predicting the submarine and sub-aerial landslide events [9]. Experimental studies and numerical modeling provide alternative ways to study landslide tsunamis [10]. For a better understanding of landslide tsunamis, many laboratory experiments were conducted. Some experiments used a solid obstacle to study the slide-type tsunamis [11; 12; 13; 14], while the others used a bulk of granular materials sliding into the water to study the slump-type landslide tsunamis [14; 15; 16]. Due to the limitation of the space, labor, and cost, the field scale experiments are rare. The difficulty and cost of the experiments make researchers adopt mathematical and numerical models to study the landslide tsunamis [17]. Because the field records are rare, the numerical models are often validated by the laboratory data [17]. However, the scale effect exists from enlarging the laboratory scale to the field scale [18; 19]. This problem is especially significant if the Reynolds number and Bingham number are involved, and the scale effect shall be validated before the applications [2]. The numerical study can be used to identify the important factors in determining wave characteristics [10]. Studies [17; 20; 21] have attempted to develop mathematical models and innovate algorithms to research the phenomenon of landslide tsunamis [22]. In recent years, the numerical models solving the shallow water equations [23; 24; 25], Boussinesq-type equation [26; 27], and Navier-Stokes equations [20; 28; 29; 30] have been proposed. In the case of a mild slope and thin thickness landslide, the shallow water approximation can be utilized on the simulation of landslide-generated tsunamis. However, for steep slopes problems, the masses slide down with high energy, in which the vertical acceleration of the landslide and ambient water cannot be neglected. In the cases of steep slopes, solving the full Navier-Stokes equations is required [31]. The landslide can be classified as block collapse (slides) and sediment collapse (slumps). In many cases, the submarine slumps are deformed [32] due to complex rheology and flow fields [33]. Deformations may change the initial acceleration of the slumps, which affects the wave generation [6; 14; 34]. The submarine slumps involve the volume of several cubic kilometers or bigger materials, including small blocks and soil, which can be modeled as a non-Newtonian fluid [35]. The deformation of the slumps should be included in the numerical model while simulating the slump-type landslide tsunamis [35]. Recently, several studies have investigated the effect of deformable slumps on wave generation [36; 37; 38; 34]. These studies described the slumps as high viscosity materials, while some studies consider the slumps as the Bingham fluids [16; 39; 40; 41]. The Bingham model has potential for describing the detailed transition processes from a solid/un-yield phase to a liquefied/yield phase [41]. Many of the studies [23; 24; 25] adopted shallow water equations as fundamental governing equations.