Advection of Inertial Particles in the Presence of the History Force: Higher Order Numerical Schemes

TL;DR

This paper develops systematic, high-order numerical schemes for accurately simulating the motion of inertial particles influenced by the history force, which involves complex integral evaluations.

Contribution

It introduces a general method to derive arbitrary order numerical schemes for the history force in inertial particle dynamics, including explicit first to third order schemes.

Findings

Schemes verified against analytical solutions.

Higher order schemes improve accuracy of inertial particle simulations.

Explicit formulas provided for practical implementation.

Abstract

The equations describing the motion of finite-size particles (inertial particles) contain in their full form the history force. This force is represented by an integral whose accurate numerical evaluation is rather difficult. Here, a systematic way is presented to derive numerical integration schemes of arbitrary order for the advection of inertial particles with the history force. This involves the numerical evaluation of integrals with singular, but integrable, integrands. Explicit specifications of first, second and third order schemes are given and the accuracy and order of the schemes are verified using known analytical solutions.