PhD Studentship: Hybrid Explicit-implicit Time Integration Schemes for Smoothed Particle Hydrodynamics

This is a 3.5 year PhD Studentship  which will cover fees and stipend set at the UKRI rate (£18,622 in 2023/24). This project is open to UK residents and EU residents who have settled status or pre-settled status.

The successful candidate will have strong mathematical background, especially in numerical analysis. Knowledge of differential equations is an advantage. Strong programming skills in C++ and Fortran are essential.

Smoothed particle hydrodynamics (SPH) is a method for numerical solution of fluid mechanics problems. In this method fluid continuum is approximated by a number of interacting particles. The position and the velocity of each particle in time is determined from the kinematic relations with the neighbouring particles. This procedure leads to the problem of solving of a system of initial value problems (IVPs).

There are two classes of methods for numerical solution of IVPs: explicit and implicit. Explicit methods are simple to implement and are computationally cheaper per time step than implicit methods. However, explicit methods suffer from instability that restricts severely the maximum allowed step size. Implicit methods are unconditionally stable, allowing to select step sizes that follow the physics of the problem under consideration, rather than method-imposed restrictions. The computational cost of implicit methods is higher than that of explicit methods and needs to be managed carefully.

The choice of a time integration scheme is inevitably problem-dependent. Many complex fluid mechanics problems exhibit multiple time scales, and the use of a single time integration scheme over the entire time interval of interest may not be the best possible choice. Instead, we should build a flexible framework that would facilitate the design of hybrid explicit-implicit time integrators for SPH. The objective of this project is to design and implement a prototype of such a system. There are two main components of this system: 1) a computational cost model for explicit and implicit methods as the basis of a decision making which type of integrator to use, and 2) the investigation of different methods for numerical estimation of the local truncation error. The system will be thoroughly tested on standard benchmarks before being deployed within the SPH method to solve more realistic problems.

We strongly recommend that you contact the supervisor(s) for this project before you apply; The Email addresses for Dr Milan Mihajlovic and Dr Georgios Fourtakas are Milan.Mihajlovic@manchester.ac.uk and Georgios.fourtakas@manchester.ac.uk

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