A High-Order Conformal FEM for Multidimensional Nonlinear Collisional Breakage Equations: Analysis and Computation

TL;DR

This paper introduces a novel high-order conformal finite element method for solving multidimensional nonlinear collisional breakage equations, achieving high accuracy and efficiency while preserving key physical quantities.

Contribution

First application of conformal FEM to solve nonlinear collisional breakage equations in multiple dimensions with proven convergence and physical quantity preservation.

Findings

Achieved high accuracy and optimal convergence rates in 1D, 2D, and 3D problems.

Preserved total count and hypervolume of particles during simulations.

Demonstrated computational efficiency in complex multidimensional scenarios.

Abstract

Particle breakage due to collisional interactions plays a vital role in the development of several phenomena in science and engineering. The nonlinear collisional breakage equations (NCBEs) are a significant set of equations in this context. Solving the NCBE is computationally challenging due to its nonlinearity, high dimensionality, and complex kernel interactions. Solving NCBE problems is more complex in two- and three-dimensional problems. In these problems, it is more challenging to evaluate multidimensional moments and integrals, maintain solution stability, and achieve computational efficiency. Despite the importance of the NCBE in science and engineering, the development of efficient numerical methods for solving it in two- and three-dimensional problems has not been adequately explored. In this work, we have introduced a new framework for solving the NCBE across multiple dimensions using the conformal finite element method (FEM). To the best of our knowledge, this is the first work to solve the NCBE using the conformal FEM. The new framework employs high-order Lagrange elements in conjunction with the BDF2 scheme for time discretization. The present method preserves the important physical quantities such as the total count and hypervolume of the population particles. Convergence results for error estimates have also been derived for both semidiscrete and fully discrete schemes. Numerical experiments have been carried out for one-, two-, and three-dimensional problems. The numerical experiments have shown that the proposed method achieved high accuracy, optimal convergence rates, and computational efficiency.