A novel approach to rigid spheroid models in viscous flows using operator splitting methods

Abstract

Calculating cost-effective solutions to particle dynamics in viscous flows is an important problem in many areas of industry and nature. We implement a second-order symmetric splitting method on the governing equations for a rigid spheroidal particle model with torques, drag and gravity. The method splits the operators into a vector field that is conservative and one that takes into account the forces of the fluid. Error analysis and numerical tests are performed on perturbed and stiff particle-fluid systems. For the perturbed case, the splitting method greatly improves the solution accuracy, when compared to a conventional multistep method, and the global error behaves as \(\mathcal {O}(\varepsilon h^{2})\) for roughly equal computational cost. For stiff systems, we show that the splitting method retains stability in regimes where conventional methods blow up. In addition, we show through numerical experiments that the global order is reduced from \(\mathcal {O}(h^{2}/\varepsilon )\) in the perturbed regime to \(\mathcal {O}(h)\) in the stiff regime.

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Funding

This work has received funding from the European Unions Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement (no. 691070) as well as the SPIRIT project (no. 231632) under the Research Council of Norway FRIPRO funding scheme. Part of this work was done while visiting the University of Cambridge, UK and La Trobe University, Melbourne, Australia.

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Correspondence to Benjamin Tapley.

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Tapley, B., Celledoni, E., Owren, B. et al. A novel approach to rigid spheroid models in viscous flows using operator splitting methods. Numer Algor 81, 1423–1441 (2019). https://doi.org/10.1007/s11075-019-00666-1

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Keywords

  • Ordinary differential equations
  • Numerical analysis
  • Splitting methods
  • Multiphase flows
  • Immersed rigid bodies
  • Order reduction