Convergence-rate and error analysis of sectional-volume average method for the collisional breakage equation with multi-dimensional modelling

TL;DR

This work analyzes a birth-modified sectional discretization, the volume-average method (VAM), for the nonlinear collisional breakage equation and its extension to two dimensions. By allocating newborn particles across adjacent cells according to the local average volume, VAM preserves mass and particle-number and is shown to be Lipschitz-stable and nonnegative. The study proves consistency and establishes a convergence framework, demonstrating first-order accuracy on uniform, nonuniform, locally-uniform, random, and oscillatory grids, with potential second-order convergence under kernel-specific conditions for some grids. Numerical experiments in 1D and 2D show VAM outperforms the fixed-pivot technique (FPT), especially on random grids, and confirm accurate preservation of key moments and hypervolume, indicating strong suitability for coupling with CFD modules.

Abstract

Recent literature reports two sectional techniques, the finite volume method [Das et al., 2020, SIAM J. Sci. Comput., 42(6): B1570-B1598] and the fixed pivot technique [Kushwah et al., 2023, Commun. Nonlinear Sci. Numer. Simul., 121(37): 107244] to solve one-dimensional collision-induced nonlinear particle breakage equation. It is observed that both the methods become inconsistent over random grids. Therefore, we propose a new birth modification strategy, where the newly born particles are proportionately allocated in three adjacent cells, depending upon the average volume in each cell. This modification technique improves the numerical model by making it consistent over random grids. A detailed convergence and error analysis for this new scheme is studied over different possible choices of grids such as uniform, nonuniform, locally-uniform, random and oscillatory grids. In addition, we have also identified the conditions upon kernels for which the convergence rate increases significantly and the scheme achieves second order of convergence over uniform, nonuniform and locally-uniform grids. The enhanced order of accuracy will enable the new model to be easily coupled with CFD-modules. Another significant advancement in the literature is done by extending the discrete model for two-dimensional equation over rectangular grids.

Figures (13)

• Figure 1: Particle allocation
• Figure 2: Particle contribution at $x_i$ from all possible cells
• Figure 3: $\bar{v}_{i}>x_{i}$ and $\bar{v}_{i}<x_{i}$
• Figure 4: Four different set of regions
• Figure 5: Uniform smooth grids

Theorems & Definitions (26)

• Remark 2.1
• Proposition 2.1
• proof
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Definition 3.4
• Definition 3.5
• Proposition 3.1
• proof