Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Finite volume convergence analysis and error estimation for non-linear collisional induced breakage equation

Sanjiv Kumar Bariwal, Rajesh Kumar

Abstract

This article focuses on the finite volume method (FVM) as an instrument tool to deal with the non-linear collisional-induced breakage equation (CBE) that arises in the particulate process. Notably, we consider the non-conservative approximation of the CBE. The analysis of weak convergence of the approximated solutions under a feasible stability condition on the time step is investigated for locally bounded breakage and collision kernels. Subsequently, explicit error estimation of the FVM solutions in uniform mesh having the kernels in the class of $W_{loc}^{1,\infty}$ space. It is also shown numerically for the first-order convergent scheme by taking numerical examples.

Finite volume convergence analysis and error estimation for non-linear collisional induced breakage equation

Abstract

This article focuses on the finite volume method (FVM) as an instrument tool to deal with the non-linear collisional-induced breakage equation (CBE) that arises in the particulate process. Notably, we consider the non-conservative approximation of the CBE. The analysis of weak convergence of the approximated solutions under a feasible stability condition on the time step is investigated for locally bounded breakage and collision kernels. Subsequently, explicit error estimation of the FVM solutions in uniform mesh having the kernels in the class of space. It is also shown numerically for the first-order convergent scheme by taking numerical examples.

Paper Structure

This paper contains 7 sections, 9 theorems, 107 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Numerical Scheme
  3. Weak Convergence
  4. Error Simulation
  5. Numerical Testing
  6. Conclusions
  7. Acknowledgments

Key Result

Theorem 3.1

Assume that the kernels hold the properties in (breakage funcn-Collisional func) and $\mathcal{C}^{in}$ satisfy (Space). Furthermore, suppose that the time step $\Delta t$ confirms the existence of a positive constant $\theta$ such that holds for Then assurance of a subsequence arises that where $\mathcal{C}$ is the weak solution of problem (maineq-initial) on $[0,\mathfrak{T}]$. This implies t

Theorems & Definitions (14)

  • Theorem 3.1
  • Theorem 3.2
  • Proposition 3.3
  • proof
  • Lemma 3.4: laurenccot2002continuous
  • Proposition 3.5
  • proof
  • Remark 3.6
  • Lemma 3.7
  • Theorem 4.1
  • ...and 4 more