On a robust ALE method with discrete primary and secondary conservation

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.