Rate of convergence and stability analysis of a modified fixed pivot technique for a fragmentation equation

This study presents the convergence and stability analysis of a recently developed fixed pivot technique for fragmentation equations (Liao et al. in Int J Numer Methods Fluids 87(4):202–215, 2018). The approach is based on preserving two integral moments of the distribution, namely (a) the zeroth-order moment, which defines the number of particles, and (b) the first-order moment, which describes the total mass in the system. The present methodology differs mathematically in a way that it delivers the total breakage rate between a mother and a daughter particle immediately, whereas existing numerical techniques provide the partial breakup rate of a mother and daughter particle. This affects the computational efficiency and makes the current model reliable for CFD simulations. The consistency and unconditional second-order convergence of the method are proved. This demonstrates efficiency of the method over the fixed pivot technique (Kumar and Warnecke in Numer Math 110(4):539–559, 2008) and the cell average technique (Kumar and Warnecke in Numer Math 111(1):81–108, 2008). Numerical results are compared against the cell average technique and the experimental order of convergence is calculated to confirm the theoretical order of convergence. </p

A Note on the Volume Conserving Solution to Simultaneous Aggregation and Collisional Breakage Equation

A new population balance model is introduced, in which a pair of particles can coagulate into a larger one if their encounter is a completely inelastic collision; otherwise, one of them breaks into multiple fragments (two or more) due to the elastic collision. Mathematically, coagulation and breakage models both manifest nonlinearity behavior. We prove the global existence and uniqueness of the solution to this model for the compactly supported kinetic kernels and an unbounded breakage distribution function. A further investigation dealt with the volume conservation property (necessary condition) of the solution

A Note on the Volume Conserving Solution to Simultaneous Aggregation and Collisional Breakage Equation

A new population balance model is introduced, in which a pair of particles can coagulate into a larger one if their encounter is a completely inelastic collision; otherwise, one of them breaks into multiple fragments (two or more) due to the elastic collision. Mathematically, coagulation and breakage models both manifest nonlinearity behavior. We prove the global existence and uniqueness of the solution to this model for the compactly supported kinetic kernels and an unbounded breakage distribution function. A further investigation dealt with the volume conservation property (necessary condition) of the solution. </p