## 5 Other Particle-Based Methods
Since the algorithm is a lot similar to that of MD, it is not schematized separately in this work. For a more detailed summary of the recent improvement, one can refer to [107]. The most important advantage of this method over MD is higher efficiency in terms of computational requirements, which makes it the preferred method in many cases because of computational tools limitation. The difference between this method and MD is in the procedure treating the interaction between the particle throughout the computational domain and this, in turn, impacts the necessary time step to ensure that the physics can be captured accurately. In MD, this time step is determined by comparing the particles' trajectories, while in DSMC, a simplified statistical equation is used to compute this time based on some assumptions. This statistical equation makes the mathematical formulation of this method simpler to understand and implement in numerical codes, since determining the time step is independent of the geometry. However, these simplifications sometimes lead to less accuracy or not trustable results in more sensitive cases, especially in cases with larger dimensions. A practical comparison of these two methods is presented in [108]. Researchers have modified this method over time to improve its accuracy over time and presented modified versions of this method to be implemented for cellular flow problems [109]. Smoothed particle hydrodynamics (SPH) is another method that has been used to investigate immersed particles in fluids over the years. This method is considered macroscopic, but particle-based and was proposed by [110]. The original method was too simplistic, but, has been improved over time and found its way for simulations of higher scale problems [111; 112]. This method is one of the oldest mesh-free methods and can model complex problems or geometries, which is one of the advantages of this method over CFD methods. The nature of this method is similar to that of DPD, in terms of the mathematical formulation and can also be compared to MD, where it is derived from second law of  
Figure 9: Simplified schematic of the MD method’s algorithm. Newton. However, this method differs from DPD or SDPD methods since it starts with NS equations and discretizes the computational domain into finite points that are treated as groups of particles using Lagrangian discretization. A good review of SPH method is presented in [113]. In addition to being a mesh-free method, unlike other particle-based methods, this method works directly with macroscopic quantities because of NS equation being the basis for the computational steps in SPH. SPH, however, is not capable of addressing the mesoscopic problems (as opposed to SDPD), where there are large thermal fluctuations introduced because of the lack of a fluctuation-dissipation equation/function. Given the premise of this method for Newtonian fluids, it is facing some issues when dealing with non-Newtonian flows. Because of its inherent drawbacks regarding mesoscale problems and mesoscale nature of RBC related problems, the application of this method has been very limited in simulation of RBCs problems. One may refer to the study conducted in [112], where the researchers studied the deformability of Malaria-infected RBCs using in house 3D code developed based on SPH method. A review comparing the applications of this method with those of MD is presented in [114]. There are other mesh-free methods available, which can especially be used for studying deformability and mechanical properties of RBCs undergoing effects of various microfluidic devices or some diseases, such as discrete particle (DP), element-free Galerkin method (EFG), moving particle semi-implicit method (MPS), moving particle finite element method (MPFEM), immersed particle method (IPM), boundary element method (BEM), and multi-particle collision dynamics method (MPCD). However, these methods have not attracted enough attention in this field, and not many studies using them can be found.