We propose a novel deterministic particle method to numerically approximate the Landau equation for plasmas. Based on a new variational formulation in terms of gradient flows of the Landau equation, we regularize the collision operator to make sense of the particle solutions. These particle solutions solve a large coupled ODE system that retains all the important properties of the Landau operator, namely the conservation of mass, momentum and energy, and the decay of entropy. We illustrate our new method by showing its performance in several test cases including the physically relevant case of the Coulomb interaction. The comparison to the exact solution and the spectral method is strikingly good maintaining 2nd order accuracy. Moreover, an efficient implementation of the method via the treecode is explored. This gives a proof of concept for the practical use of our method when coupled with the classical PIC method for the Vlasov equation.
Bibliographical noteFunding Information:
JAC, JH and LW would like to thank the American Institute of Mathematics for their support through a SQuaREs project where this work was finished. This research was generated from the AIM workshop “Nonlocal differential equations in collective behavior” in June 2018. JH would like to thank Tong Ding for testing parameters in an undergraduate research project related to the current work. LW would like to thank Prof. Robert Krasny for fruitful discussion on treecode and Dr. Evan Bollig on the help with Minnesota super computers. JAC was partially supported by EPSRC grant number EP/P031587/1 and the Advanced Grant Nonlocal-CPD (Nonlocal PDEs for Complex Particle Dynamics: Phase Transitions, Patterns and Synchronization) of the European Research Council Executive Agency (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 883363 ). JH was partially supported by NSF DMS-1620250 and NSF CAREER grant DMS-1654152 . LW was partially supported by NSF DMS-1903420 and NSF CAREER grant DMS-1846854 . JW was funded by the President's PhD Scholarship program of Imperial College London .
- Deterministic particle methods
- Gradient flows
- Landau equation for plasmas