Applications Of Structure Preserving Algoritms In The Plasma Physics

Hamiltonian system is very commonly used for describing physical processes.Its evolution conserves a so-called symplectic structure,which guarantees the conservation of invariants and the long-term stability of the evolution.These con-servation properties will be helpful when we are discussing and understanding the long-term property of the physical process.There are four fundamental models in the plasma physics,e.g.test particle,collisionless Vlasov-Maxwell,ideal two-fluid and ideal magnetohydrodynamics,and they are all Hamiltonian systems.Devel-oping algorithms for solving these fundamental models efficiently are very useful and important for us to investigate phenomena in these models.However,traditional algorithms based on discretizing the differential equations directly usually breaks the conservation property of these Hamiltonian systems and the numeric error will keep accumulating when doing the time iteration.This will become a serious problem when facing multiple time-scale problems which need a lot time steps to resolve all high frequency physics.In this paper,we start from the theory of symplectic algorithms,give an introduction to symplectic algorithms and how to construct them,and construct symplectic algorithms for all 4 fundamental models of the plasma physics.Some numerical examples are given to verify the correctness and long-term property of these algorithms.For particle dynamics,there are already symplectic algorithms because the test-particle system is a finite dimensional canonical Hamiltonian system.Firstly,we constructed variational integrators for both non-relativistic and relativistic par-ticles in external electromagnetic fields.Then we created gauge-free non-canonical symplectic algorithms for these two test-particle models.Finally we chose the dynamics of a charged particle in the Tokamak geometry as an example to test the correctness and long-term conservative property of these algorithms.Vlasov-Maxwell model is a model which describes the plasma using electro-magnetic fields and particle distribution functions.The Vlasov-Maxwell system is very similar to the original particle-field plasma system.However,it is a infinite dimensional non-canonical Hamiltonian system and generally speaking the cor-responding symplectic algorithms is hard to obtain.In practice people often use particles to discretize distribution functions in the Vlasov-Maxwell system,and the corresponding method is called particle-in-cell method.Firstly,we discretize the Lagrangian of the particle-field system and obtain a 1st-order variational particle-in-cell scheme.Then to construct gauge-free schemes,we created a multi-grid Whitney interpolation form for cubic meshes.With this interpolation form,discrete exterior calculus and Hamiltonian splitting method we have constructed ahigh-order charge-conservative non-canonical symplectic particle-in-cell scheme.Symplectic particle-in-cell methods for relativistic plasmas was also proposed.Totest these methods,two numerical examples are chosen,which are the dispersion relation of X-Bernstein waves and the Landau damping.Ideal two-fluid system is also an infinite dimensional non-canonical Hamil-tonian system.In this model particles are treated as charged fluids and evolved with electromagnetic fields.We used a similar idea as the symplectic algorithms for Vlasov-Maxwell systems to construct the high-order charge-conservative non-canonical symplectic two-fluid scheme.In this method,the multi-grid Whitney interpolation form for cubic meshes,discrete exterior calculus and the Hamilto-nian splitting method are used to ensure the gauge-free property and high-order explicitness.Two numerical examples are chosen to verify this method,which are the dispersion relation of a magnetized two-fluid plasma and the two-streaminstability.Ideal magnetohydrodynamics is a reduced model of plasmas.In this model,high-frequency electron motions are neglected to simplify the problen when mod-eling low frequency plasmas.It is a non-canonical Hamiltonian system more com-plex than the two-fluid system.This is because the evolution not only conservesthe non-canonical symplectic structure,but also conserves a topological structure of magnetic field(i.e.flux freezing).We start from the variational principle of the magnetohydrodynamics in the Euleian description,discretize the Lagrangian with constraints and use the discrete variational principle to obtain the final symplectic magnetohydrodynamics algorithm.The conservation property of this algorithmis tested by a numeric example,which is the dispersion relation of the magnetohydrodynamics system.Symplectic algorithms displayed in this paper are essentially structure-preserving approximations of original Hamiltonian plasma systems.According to the theory of symplectic algorithms,these discrete systems are all Hamiltonian systems,thus their long-term behaviors are much better than conventional methods.These excellent properties are helpful for us to simulate and predict behaviors of plasmas,and then have a better understanding of the complex physics in plasmas.