Convergence of particle methods, particularly SPH

• Applicant: Professor Dr. Holger Wendland
• Subject Area: Mathematics
• Term: from 2014 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 262166066

Particle methods represent an attractive and modern method for the numerical simulation of transport problems. Particle methods are, in contrast to classical methods like Finite Differences, Finite Elements and Finite Volumes, based upon a Lagrangian formulation of the problem. Hence, they are particularly useful for advection dominated flows and for problems with a varying topology. In recent years, amongst all particle methods, a technique called SPH (smoothed particle hydrodynamics) has become extremely popular. SPH is a mesh-free, kernel-based approximation method, which is, for example, already employed in such diverse applications as astrophysics, geotechnical and environmental engineering, hydraulic engineering, harbour construction and solid-state physics. Despite some attempts, so far, there is no rigorous mathematical treatment of the SPH method, particularly when it comes to applications in fluid dynamics. Hence, it is the goal of this project to provide a thorough error analysis of the Euler equations for inviscid fluid flow. For the best possible flexibility regarding the choice of the time discretisation, we will first only look at the spacial discretisation with SPH and analyse the semi-discrete scheme resulting from this. After that, we will analyse some explicit time-discretisation schemes. Since there are several versions of SPH to discretise spatial derivatives, we will examine and compare the most popular ones amongst them. Finally, in a second step, we will analyse the SPH discretisation of the Navier-Stokes equations for viscous fluids. To this end, the error estimates derived thus far within this project have to be extended to also cover the discretisation of the Laplace operator. In all these cases, we are interested in proving error estimates with particular emphasis on the connection between the different discretisation parameters. So far, these parameters are only chosen problem dependent and heuristically. A precise mathematically established connection between the discretisation parameters will therefore also lead to a significant improvement when it comes to numerical simulation.

• DFG Programme: Research Grants