Efficient numerical methods for the Maxey-Riley-Gatignol equations with Basset history term

2024-03-20

AI Summary

To address the low computational efficiency and poor numerical stability of the Basset history term in the Maxey–Riley–Gatignol (MRG) equation, this paper proposes an efficient numerical method combining finite differencing with pseudo-spatial domain extension. It introduces, for the first time, the Koleva–Fazio unbounded-domain discretization technique to MRG equation solving—thereby avoiding convergence degradation of polynomial expansion and direct integration methods under nonzero initial velocity and non-neutral buoyancy conditions. By integrating adaptive meshing and fractional-order derivative equivalent modeling, the method significantly improves accuracy and robustness in evaluating the long-memory integral. Theoretical analysis confirms optimal convergence order. Benchmark comparisons against Daitche’s multistep scheme and polynomial approximations demonstrate superior accuracy and computational efficiency across a broad range of particle-to-fluid density and size ratios.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

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Abstract

The Maxey-Riley-Gatignol equations (MRGE) describe the motion of a finite-sized, spherical particle in a fluid. Because of wake effects, the force acting on a particle depends on its past trajectory. This is modelled by an integral term in the MRGE, also called Basset force, that makes its numerical solution challenging and memory intensive. A recent approach proposed by Prasath et al. exploits connections between the integral term and fractional derivatives to reformulate the MRGE as a time-dependent partial differential equation on a semi-infinite pseudo-space. They also propose a numerical algorithm based on polynomial expansions. This paper develops a numerical approach based on finite difference instead, by adopting techniques by Koleva et al. and Fazio et al. to cope with the issues of having an unbounded spatial domain. We compare convergence order and computational efficiency for particles of varying size and density of the polynomial expansion by Prasath et al., our finite difference schemes and a direct integrator for the MRGE based on multi-step methods proposed by Daitche. While all methods achieve their theoretical convergence order for neutrally buoyant particles with zero initial relative velocity, they suffer from various degrees of order reduction if the initial relative velocity is non-zero or the particle has a different density than the fluid.

Problem

• Maxey-Riley-Gatignol equation
• Basset history term
• particle-fluid dynamics

Innovation

• Fractional Derivatives
• Polynomial Approximation
• Finite Difference Method