Explicit integrators for nonlocal equations: The case of the Maxey-Riley-Gatignol equation
Authors:
Divya Jaganathan, Rama Govindarajan and Vishal Vasan
Journal:
Quart. Appl. Math.
MSC (2020):
Primary 65R20, 45J05; Secondary 65L06, 76D07
DOI:
https://doi.org/10.1090/qam/1693
Published electronically:
March 25, 2024
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Abstract: The Maxey-Riley-Gatignol (MRG) equation, which describes the dynamics of an inertial particle in nonuniform and unsteady flow, is an integro-differential equation with a memory term and its solution lacks a well-defined Taylor series at $t=0$. In particulate flows, one often seeks trajectories of millions of particles simultaneously, and the numerical solution to the MRG equation for each particle becomes prohibitively expensive due to its ever-rising memory costs. In this paper, we present an explicit numerical integrator for the MRG equation that inherits the benefits of standard time-integrators, namely a constant memory storage cost, a linear growth of operational effort with simulation time, and the ability to restart a simulation with the final state as the new initial condition. The integrator is based on a Markovian embedding of the MRG equation. The integrator and the embedding are consequences of a spectral representation of the solution to the linear MRG equation. We exploit these to extend the work of Cox and Matthews [J. Comput. Phys. 176 (2002), 430–455] and derive Runge-Kutta type iterative schemes of differing orders for the MRG equation. Our approach may be generalized to a large class of systems with memory effects.
References
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- P. A. Moreno-Casas and F. A. Bombardelli, Computation of the Basset force: recent advances and environmental flow applications, Environ. Fluid Mech. 16 (2016), 193–208.
- M. Parmar, S. Annamalai, S. Balachandar, and A. Prosperetti, Differential formulation of the viscous history force on a particle for efficient and accurate computation, J. Fluid Mech. 844 (2018), 970–993. MR 3787410, DOI 10.1017/jfm.2018.217
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- P. Siegle, I. Goychuk, and P. Hänggi, Markovian embedding of fractional superdiffusion, Europhys. Lett. 93 (2011), no. 2, 20002.
- M. A. T. van Hinsberg, J. H. M. ten Thije Boonkkamp, and H. J. H. Clercx, An efficient, second order method for the approximation of the Basset history force, J. Comput. Phys. 230 (2011), no. 4, 1465–1478.
References
- T. R. Auton, J. C. R. Hunt, and M. Prud’homme, The force exerted on a body in inviscid unsteady nonuniform rotational flow, J. Fluid Mech. 197 (1988), 241–257. MR 1018039, DOI 10.1017/S0022112088003246
- A. Basset, Treatise on hydrodynamics, Deighton, Bell and Company, 1888.
- F. Bombardelli, A. González, and Y. Niño, Computation of the particle Basset force with a fractional-derivative approach, J. Hydraul. Eng. 134 (2008), no. 10, 1513–1520.
- J. Boussinesq, Sur la résistance qu’oppose un liquide indéfini au repos au mouvement varié d’une sphère solide, C. R. Acad. Sci. Paris 100 (1885), 935–937.
- L. M. Brush, H-W. Ho, and B-C. Yen, Accelerated motion of a sphere in a viscous fluid, J. Hydraul. Eng. 90 (1964), 149–160.
- S. Campbell, F. Ciccarello, G. M. Palma, and B. Vacchini, System-environment correlations and Markovian embedding of quantum non-Markovian dynamics, Phys. Rev. A 98 (2018), 012142.
- S. M. Cox and P. C. Matthews, Exponential time differencing for stiff systems, J. Comput. Phys. 176 (2002), no. 2, 430–455. MR 1894772, DOI 10.1006/jcph.2002.6995
- Ruth F. Curtain and Hans Zwart, An introduction to infinite-dimensional linear systems theory, Texts in Applied Mathematics, vol. 21, Springer-Verlag, New York, 1995. MR 1351248, DOI 10.1007/978-1-4612-4224-6
- Anton Daitche, Advection of inertial particles in the presence of the history force: higher order numerical schemes, J. Comput. Phys. 254 (2013), 93–106. MR 3143359, DOI 10.1016/j.jcp.2013.07.024
- A. J. Dorgan and E. Loth, Efficient calculation of the history force at finite Reynolds numbers, Int. J. Multiphase Flow 33 (2007), no. 8, 833–848.
- Mohammad Farazmand and George Haller, The Maxey-Riley equation: existence, uniqueness and regularity of solutions, Nonlinear Anal. Real World Appl. 22 (2015), 98–106. MR 3280821, DOI 10.1016/j.nonrwa.2014.08.002
- Roberto Garrappa, Stability-preserving high-order methods for multiterm fractional differential equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 22 (2012), no. 4, 1250073, 13. MR 2926049, DOI 10.1142/S0218127412500733
- Roberto Garrappa and Marina Popolizio, Generalized exponential time differencing methods for fractional order problems, Comput. Math. Appl. 62 (2011), no. 3, 876–890. MR 2824677, DOI 10.1016/j.camwa.2011.04.054
- R. Gatignol, The Faxén formulas for a rigid particle in an unsteady non-uniform Stokes-flow, J. Mec. Theor. Appl. 2 (1983), 143–160.
- Marlis Hochbruck and Alexander Ostermann, Explicit exponential Runge-Kutta methods for semilinear parabolic problems, SIAM J. Numer. Anal. 43 (2005), no. 3, 1069–1090. MR 2177796, DOI 10.1137/040611434
- D. Jaganathan, S. G. Prasath, R. Govindarajan, and V. Vasan, The Basset–Boussinesq history force: its neglect, validity, and recent numerical developments, Front. Phys. 11 (2023), 370.
- S. Kretschmer, K. Luoma, and W. T. Strunz, Collision model for non-Markovian quantum dynamics, Phys. Rev. A 94 (2016), 012106.
- Gabriel Provencher Langlois, Mohammad Farazmand, and George Haller, Asymptotic dynamics of inertial particles with memory, J. Nonlinear Sci. 25 (2015), no. 6, 1225–1255. MR 3415046, DOI 10.1007/s00332-015-9250-0
- Phillip M. Lovalenti and John F. Brady, The hydrodynamic force on a rigid particle undergoing arbitrary time-dependent motion at small Reynolds number, J. Fluid Mech. 256 (1993), 561–605. MR 1251413, DOI 10.1017/S0022112093002885
- M. R. Maxey and J. J. Riley, Equation of motion for a small rigid sphere in a nonuniform flow, Phys. Fluids 26 (1983), no. 4, 883–889.
- R. McCloskey and M. Paternostro, Non-Markovianity and system-environment correlations in a microscopic collision model, Phys. Rev. A 89 (2014), 052120.
- P. A. Moreno-Casas and F. A. Bombardelli, Computation of the Basset force: recent advances and environmental flow applications, Environ. Fluid Mech. 16 (2016), 193–208.
- M. Parmar, S. Annamalai, S. Balachandar, and A. Prosperetti, Differential formulation of the viscous history force on a particle for efficient and accurate computation, J. Fluid Mech. 844 (2018), 970–993. MR 3787410, DOI 10.1017/jfm.2018.217
- S. Ganga Prasath, Vishal Vasan, and Rama Govindarajan, Accurate solution method for the Maxey-Riley equation, and the effects of Basset history, J. Fluid Mech. 868 (2019), 428–460. MR 3938307, DOI 10.1017/jfm.2019.194
- P. Siegle, I. Goychuk, and P. Hänggi, Markovian embedding of fractional superdiffusion, Europhys. Lett. 93 (2011), no. 2, 20002.
- M. A. T. van Hinsberg, J. H. M. ten Thije Boonkkamp, and H. J. H. Clercx, An efficient, second order method for the approximation of the Basset history force, J. Comput. Phys. 230 (2011), no. 4, 1465–1478.
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Additional Information
Divya Jaganathan
Affiliation:
International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
ORCID:
0009-0008-7107-5655
Email:
divya.jaganathan@icts.res.in
Rama Govindarajan
Affiliation:
International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
MR Author ID:
600086
ORCID:
0009-0007-4794-4964
Email:
rama@icts.res.in
Vishal Vasan
Affiliation:
International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
MR Author ID:
992275
ORCID:
0000-0001-6588-6461
Email:
vishal.vasan@icts.res.in
Received by editor(s):
September 13, 2023
Received by editor(s) in revised form:
February 3, 2024
Published electronically:
March 25, 2024
Additional Notes:
Research at TIFR is supported by the Department of Atomic Energy, Government of India, under Project No. RTI4001. The first author acknowledges the hospitality at KITP, Santa Barbara during the program Multiphase Flows in Geophysics and the Environment (2022) which was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958. The third author was supported through the SERB MATRICS Grant (MTR/2019/000609) from the Science and Engineering Research Board (SERB), Department of Science and Technology, Government of India.
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