Explicit symplectic algorithms based on generating functions for charged particle dynamics

Dynamics of a charged particle in the canonical coordinates is a Hamiltonian system, and the well-known symplectic algorithm has been regarded as the de facto method for numerical integration of Hamiltonian systems due to its long-term accuracy and fidelity. For long-term simulations with high efficiency, explicit symplectic algorithms are desirable. However, it is generally believed that explicit symplectic algorithms are only available for sum-separable Hamiltonians, and this restriction limits the application of explicit symplectic algorithms to charged particle dynamics. To overcome this difficulty, we combine the familiar sum-split method and a generating function method to construct second- and third-order explicit symplectic algorithms for dynamics of charged particle. The generating function method is designed to generate explicit symplectic algorithms for product-separable Hamiltonian with form of $H(\mathbf{x},\mathbf{p})={\mathbf{p}}_{i}f\left(\mathbf{x}\right)$ or $H(\mathbf{x},\mathbf{p})={\mathbf{x}}_{i}g\left(\mathbf{p}\right)$. Applied to the simulations of charged particle dynamics, the explicit symplectic algorithms based on generating functions demonstrate superiorities in conservation and efficiency.

Figure 1

Convergence rate of the energy error for four symplectic methods. It verifies that ${\mathrm{\Psi }}_{\mathrm{\Delta }t}^2$ is indeed a second-order method and ${\mathrm{\Psi }}_{\mathrm{\Delta }t}^3$ is a third-order method.

Figure 2

Two-dimensional tokamak geometry with circular concentric flux surfaces.

Figure 3

Simulations of long-term dynamics of a $3.5\mathrm{MeV}\alpha$ particle in a tokamak. The initial orbits are plotted using blue lines, and the orbits after $1.998\times {10}^7$ steps are plotted using red lines. (a) Numerical orbit obtained by a nonsymplectic RK4 method. (b) The orbit obtained by the IS2 method. (c) The numerical orbit by the ES2 method ${\mathrm{\Psi }}_{\mathrm{\Delta }t}^2$. (d) The normalized energy $H/H_0$ of three methods are plotted as functions of simulation time step.