[Paper Review] On the numerical solution of a variable-coefficient Burgers equation arising in granular segregation

Ivan C. Christov|arXiv (Cornell University)|Jun 30, 2017

Granular flow and fluidized beds2 references3 citations

TL;DR

This paper presents a second-order accurate, strongly implicit Crank-Nicolson numerical scheme for solving a variable-coefficient Burgers equation modeling granular segregation in dry bidisperse mixtures. The method handles nonlinear boundary conditions for particle flux and is validated against an exact kink solution, demonstrating second-order convergence and good conservation properties, with applications to both constant- and variable-coefficient cases based on simulation-derived kinetic stress profiles.

ABSTRACT

We study a variable-coefficient Burgers equation arising in the modelling of segregation of dry bidisperse granular mixtures. The equation is subject to nonlinear boundary conditions for the particle flux. We construct a strongly implicit Crank--Nicolson type of numerical scheme for the latter equation. The scheme is benchmarked against a standard exact solution of kink type, showing second-order of accuracy and good discrete conservation properties. Two segregation problems considered in the literature are then solved and discussed. The first is the case of a linear kinetic stress profile, which renders the governing equation of constant-coefficient type, while the second is the case of a variable kinetic stress profile based on an expression fit to particle dynamics simulation data.

Motivation & Objective

• To develop a robust numerical scheme for a variable-coefficient Burgers equation arising in granular segregation of bidisperse mixtures.
• To address nonlinear boundary conditions for particle flux in the governing equation.
• To ensure second-order accuracy and discrete conservation properties in the numerical solution.
• To benchmark the scheme against an exact kink-type solution for validation.
• To apply the scheme to two physical cases: linear and variable kinetic stress profiles based on particle dynamics simulations.

Proposed method

• A strongly implicit Crank-Nicolson-type finite difference scheme is constructed for the variable-coefficient Burgers equation.
• The scheme is formulated to handle nonlinear boundary conditions on particle flux.
• The method is validated using an exact kink-type analytical solution, confirming second-order convergence.
• The scheme preserves discrete conservation properties, critical for physical consistency.
• Two physical scenarios are simulated: one with a linear kinetic stress profile (constant-coefficient case), and another with a variable profile derived from particle dynamics simulations.
• The numerical results are compared with existing literature to assess accuracy and physical relevance.

Experimental results

Research questions

• RQ1 How can a numerical scheme be designed to accurately solve a variable-coefficient Burgers equation with nonlinear flux boundary conditions in granular segregation?
• RQ2 What is the convergence order and conservation behavior of the proposed numerical scheme?
• RQ3 How does the model perform when applied to a constant-coefficient case with a linear kinetic stress profile?
• RQ4 How does the model behave under a variable kinetic stress profile derived from particle dynamics simulations?
• RQ5 What insights into segregation dynamics can be gained from solving the full variable-coefficient equation numerically?

Key findings

• The proposed Crank-Nicolson scheme achieves second-order accuracy in both space and time, as confirmed by convergence testing against an exact kink solution.
• The scheme exhibits good discrete conservation properties, preserving physical invariants crucial for modeling particle flux.
• For the linear kinetic stress case, the solution matches the known constant-coefficient Burgers equation behavior.
• The variable-coefficient case, based on a fit to particle dynamics simulation data, reveals distinct segregation dynamics not captured by constant-coefficient models.
• The numerical results are consistent with prior literature, validating the model's physical relevance and the scheme's reliability.