# An efficient numerical method for the Maxey-Riley Equation

### Working Groups: Chair Computational Mathematics

### Collaborators (MAT): Julio Urizarna, M. Sc., Prof.Â Dr.Â Daniel Ruprecht

### Collaborators (External): Prof.Â Dr.Â Kathrin Padberg-Gehle, Prof.Â Dr.Â Alexandra von Kameke

## Description

The Maxey-Riley Equation (MRE) models the motion of a finite-sized spherical particle moving in a fluid. Applications using the MRE are, for example, the study of the spread of Coronavirus particles in a room, the formation of clouds and the so-called marine snow. The MRE is a second-order, implicit integro-differential equation with a singular kernel at initial time. For over 35 years, researchers used approximations and numerical schemes with high storage requirements or ignored the integral term, although its impact can be relevant. A major break-through was reached in 2019, when Prasath et al.Â mapped the MRE to a time-dependent Robin-type bounday condition of the 1D Heat Equation, thus removing the requirement to store the full history. They provided an implicit integral form of the solution by using the so-called Fokas method that could be later solved with a numerical scheme and a nonlinear solver.

Prasath et al.â€™s research was presented without any accompanying code or program, therefore, we decided to implement an open Python ToolBox that included their theory and that could be used to solve the MRE in both theoretical and experimental flows. This work has already been finalised and accepted for publication in the journal *Proceedings for Applied Mathematics and Mechanics*.

Additionally, we are working on finding new efficient approaches to solve the MRE, since even though Prasath et al.â€™s method can deliver numerical solutions of very high accuracy, the need to evaluate nested integrals, is a process that makes it computationally costly and it becomes impractical for computing trajectories of a large number of particles.