Team/People

My research is focused on the development of structure-preserving and efficient numerical methods to solve the high-dimensional Vlasov equation. The Vlasov equation arises from kinetic theory and is required if the assumption of the velocity distribution being close to equilibrium is no longer satisfied and therefore fluid-models are no longer applicable. For plasmas this is often the case for rarefied or very hot plasmas, which occur in regions of fusion reactors or are relevant for the simulation of electric space propulsion devices.

The Vlasov equation involves solving a system of up to seven-dimensional (1 time, 3 space, 3 velocity) linear transport equations, which are non-linearly coupled together through self-induced electro-magnetic forces computed through the Maxwell's equations. Solving this system by classical means as e.g. a Finite Volume or Finite Element method is for most cases prohibitively expensive due to the curse-of-dimensionality. Therefore the most common approaches for this type of problem involve a particle-discretization as in e.g. Particle-In-Cell (PIC), which allow for simpler distributed computing with reasonably good scaling, though still suffering from high memory-footprint. However, in contrast to purely Eulerian schemes, particle-based schemes tend to suffer from numerical/statistical noise making them somewhat unrealiable.

In my work I'm trying to develop numerical schemes which try tackling both the curse of dimensionality and noise-problem through the Lagrangian framework. To this end we developed a scheme, the Numerical Flow Iteration, showing that it is sufficient to store the evolution of the (self-induced) electro-magnetic fields over time and then it is possible to accurately evaluate the individual distribution functions without having to explicitly storing them. This allows for higher accuracy while also reducing the memory requirement of Vlasov simulations by several orders of magnitude at the cost of a higher computational complexity (quadratic instead of linear in time). The respective code can be found in my GitHub repository: NuFI.

As alternative to the classical PIC approach for the Vlasov-Poisson system we also worked on kernel-interpolation-based particle methods. This allows for a lower numerical noise ratio in Vlasov simulations, but comes at the cost of a more expensive advection step as a large (potentially ill-conditioned) linear system has to be solved in each time-step. The respective code base can also be found in my GitHub respository: vlasovius.

Education

• 10/2014 - 09/2017 B.Sc. in Mathematics at RWTH Aachen
• 10/2017 - 10/2019 M.Sc. in Mathematics at RWTH Aachen
• 04/2020-09/2024 PhD candidate at ACoM, RWTH Aachen
• since 10/2024 Post-doctoral researcher at ACoM, RWTH Aachen

Research Interests

• Particle methods
• Kinetic theory in context of Plasma physics
• Reproducing kernel Hilbert spaces (RKHS)
• Numerical Analysis
• High-Performance Computing
• Continuous Optimization theory and differential geometry

Scholarships

• Part of the NHR graduate school since April 2022

Selected Conference Contributions and Invited Talks

• NHR conference, Darmstadt, Germany, 2024: Towards using the Numerical Flow Iteration to simulate kinetic plasma physics
• RGD, Goettingen, Germany, 2024: Simulation of multi-species kinetic turbulences with the Numerical Flow Iteration
• IMSI Computational Challenges and Optimization in Kinetic Plasma Physics, Chicago, USA, 2024: Discussion of potentials and draw-backs of using the Numerical Flow Iteration to solve the Vlasov equation
• HPC Asia, Nagoya, Japan, 2024': Performance comparison of the Numerical Flow Iteration to Lagrangian and Semi-Lagrangian approaches for solving the Vlasov equation in the six-dimensional phase-space'
• NHR conference, Berlin, Germany, 2023: Comparison of the Numerical Flow Iteration to particle-based approaches for the Vlasov equation
• GAMM, Dresden, Germany, 2023: The numerical flow iteration for the Vlasov equation
• Research Visit University Bochum, 2023: NuFI: An implicit Lagrangian Flow reconstruction scheme for the Vlasov equation
• SIAM CSE, Amsterdam, Netherlands, 2023: NuFI: The numerical flow iteration for the Vlasov-Poisson equation
• Research Visit TU Darmstadt, 2023: NuFI: The numerical flow iteration for the Vlasov-Poisson equation
• HPC Fusion, Barcelona, Spain (virtual), 2022: NuFI: The numerical flow iteration for the Vlasov-Poisson equation
• WCCM-APCOM, Yokohama, Japan (virtual), 2022: An interpolating particle method for the Vlasov-Poisson equation

Publications and Preprints

• Rostislav-Paul Wilhelm, 'Structure preserving numerical methods for the Vlasov equation without a phase-space mesh', PhD thesis at RWTH Aachen University (2024).

• Rostislav-Paul Wilhelm, Jan Eifert, Manuel Torrilhon, Fabian Orland, 'High fidelity simulations of the multi-species Vlasov equation in the electro-static, collision-less limit', submitted to Special Issue on High Performance Supercomputing (HPC) in Fusion Research 2023 in Plasma Physics and Controlled Fusion (2024).

• Rostislav-Paul Wilhelm, Matthias Kirchhart and Manuel Torrilhon. ‘Introduction to the numerical flow iteration for the Vlasov–Poisson equation’. Proceedings in Applied Mathematics and Mechanics, 10.1002/pamm.202300162 (Sep. 2023).

• Matthias Kirchhart and Rostislav-Paul Wilhelm. ‘The Numerical Flow Iteration for the Vlasov–Poisson equation’. SIAM Journal on Scientific Computing, Vol. 46.3, 10.1137/23M154710X (2024).

• Rostislav-Paul Wilhelm and Matthias Kirchhart. ‘An interpolating particle method for the Vlasov–Poisson equation’. Journal of computational Physics, 10.1016/j.jcp.2022.111720 (Jan. 2023).

• Rostislav-Paul Wilhelm and Michael Westdickenberg. 'Existence of gradient flows in Riemannian geometry of matrix spaces induced by the Bogoliubov inner product', 10.13140/RG.2.2.17231.84640 (Sep. 2019).