Title: Discontinuous Galerkin schemes for a class of Hamiltonian evolution equations with applications to plasma fluid and kinetic problems Show Chinese title

Title: 用于一类哈密顿演化方程的不连续伽辽金格式及其在等离子体流体和动力学问题中的应用

Abstract: In this paper we present energy-conserving, mixed discontinuous Galerkin (DG) and continuous Galerkin (CG) schemes for the solution of a broad class of physical systems described by Hamiltonian evolution equations. These systems often arise in fluid mechanics (incompressible Euler equations) and plasma physics (Vlasov--Poisson equations and gyrokinetic equations), for example. The dynamics is described by a distribution function that evolves given a Hamiltonian and a corresponding Poisson bracket operator, with the Hamiltonian itself computed from field equations. Hamiltonian systems have several conserved quantities, including the quadratic invariants of total energy and the $L_2$ norm of the distribution function. For accurate simulations one must ensure that these quadratic invariants are conserved by the discrete scheme. We show that using a discontinuous Galerkin scheme to evolve the distribution function and ensuring that the Hamiltonian lies in its continuous subspace leads to an energy-conserving scheme in the continuous-time limit. Further, the $L_2$ norm is conserved if central fluxes are used to update the distribution function, but decays monotonically when using upwind fluxes. The conservation of density and $L_2$ norm is then used to show that the entropy is a non-decreasing function of time. The proofs shown here apply to any Hamiltonian system, including ones in which the Poisson bracket operator is non-canonical (for example, the gyrokinetic equations). We demonstrate the ability of the scheme to solve the Vlasov--Poisson and incompressible Euler equations in 2D and provide references where we have applied these schemes to solve the much more complex 5D electrostatic and electromagnetic gyrokinetic equations. Show Chinese abstract

Abstract: 本文我们提出了能量守恒的混合不连续伽辽金（DG）和连续伽辽金（CG）方案，用于求解由哈密顿演化方程描述的广泛物理系统。 这些系统通常出现在流体力学（不可压缩欧拉方程）和等离子体物理（Vlasov--Poisson方程和gyrokinetic方程）中。 动力学由一个分布函数描述，在给定哈密顿量和相应的泊松括号算子的情况下演化，而哈密顿量本身则由场方程计算得出。 哈密顿系统有几个守恒量，包括总能量的二次不变量和分布函数的$L_2$范数。 为了准确模拟，必须确保这些二次不变量由离散方案守恒。 我们证明，使用不连续伽辽金方案演化分布函数，并确保哈密顿量位于其连续子空间中，会导致连续时间极限下的能量守恒方案。 此外，当使用中心通量更新分布函数时，$L_2$范数被守恒，但使用迎风通量时则单调衰减。 随后利用密度和$L_2$范数的守恒性证明熵是时间的非递减函数。 此处展示的证明适用于任何哈密顿系统，包括泊松括号算子为非规范的情况（例如，gyrokinetic方程）。 我们展示了该方案求解二维Vlasov--Poisson和不可压缩欧拉方程的能力，并提供了参考文献，说明我们已将这些方案应用于求解更复杂的五维静电和电磁gyrokinetic方程。