The classical Pauli particle(CPP) serves as a slow manifold, substituting the conventional guiding center dynamics. Based on the CPP, we utilize the averaged vector field(AVF) method in the computations of drift orbit...The classical Pauli particle(CPP) serves as a slow manifold, substituting the conventional guiding center dynamics. Based on the CPP, we utilize the averaged vector field(AVF) method in the computations of drift orbits. Demonstrating significantly higher efficiency, this advanced method is capable of accomplishing the simulation in less than one-third of the time of directly computing the guiding center motion. In contrast to the CPP-based Boris algorithm, this approach inherits the advantages of the AVF method, yielding stable trajectories even achieved with a tenfold time step and reducing the energy error by two orders of magnitude. By comparing these two CPP algorithms with the traditional RK4 method, the numerical results indicate a remarkable performance in terms of both the computational efficiency and error elimination. Moreover, we verify the properties of slow manifold integrators and successfully observe the bounce on both sides of the limiting slow manifold with deliberately chosen perturbed initial conditions. To evaluate the practical value of the methods, we conduct simulations in non-axisymmetric perturbation magnetic fields as part of the experiments,demonstrating that our CPP-based AVF method can handle simulations under complex magnetic field configurations with high accuracy, which the CPP-based Boris algorithm lacks. Through numerical experiments, we demonstrate that the CPP can replace guiding center dynamics in using energy-preserving algorithms for computations, providing a new, efficient, as well as stable approach for applying structure-preserving algorithms in plasma simulations.展开更多
In this paper, the dissipative and the forced terms of the Duffing equation are considered as the perturbations of nonlinear Hamiltonian equations and the perturbational effect is indicated by parameter ε. Firstly, b...In this paper, the dissipative and the forced terms of the Duffing equation are considered as the perturbations of nonlinear Hamiltonian equations and the perturbational effect is indicated by parameter ε. Firstly, based on the gradient- Hamiltonian decomposition theory of vector fields, by using splitting methods, this paper constructs structure-preserving algorithms （SPAs） for the Duffing equation. Then, according to the Liouville formula, it proves that the Jacobian matrix determinants of the SPAs are equal to that of the exact flow of the Duffing equation. However, considering the explicit Runge Kutta methods, this paper finds that there is an error term of order p＋l for the Jacobian matrix determinants. The volume evolution law of a given region in phase space is discussed for different algorithms, respectively. As a result, the sum of Lyapunov exponents is exactly invariable for the SPAs proposed in this paper. Finally, through numerical experiments, relative norm errors and absolute energy errors of phase trajectories of the SPAs and the Heun method （a second-order Runge-Kutta method） are compared. Computational results illustrate that the SPAs are evidently better than the Heun method when e is small or equal to zero.展开更多
Recent development of structure-preserving geometric particle-in-cell （PIC） algorithms for Vlasov-Maxwell systems is summarized. With the arrival of 100 petaflop and exaflop computing power, it is now possible to ca...Recent development of structure-preserving geometric particle-in-cell （PIC） algorithms for Vlasov-Maxwell systems is summarized. With the arrival of 100 petaflop and exaflop computing power, it is now possible to carry out direct simulations of multi-scale plasma dynamics based on first-principles. However, standard algorithms currently adopted by the plasma physics community do not possess the long-term accuracy and fidelity required for these large-scale simulations. This is because conventional simulation algorithms are based on numerically solving the underpinning differential （or integro-differential） equations, and the algorithms used in general do not preserve the geometric and physical structures of the systems, such as the local energy-momentum conservation law, the symplectic structure, and the gauge symmetry. As a consequence, numerical errors accumulate coherently with time and long-term simulation results are not reliable. To overcome this difficulty and to harness the power of exascale computers, a new generation of structure-preserving geometric PIC algorithms have been developed. This new generation of algorithms utilizes modem mathematical techniques, such as discrete manifolds, interpolating differential forms, and non-canonical symplectic integrators, to ensure gauge symmetry, space-time symmetry and the conservation of charge, energy-momentum, and the symplectic structure. These highly desired properties are difficult to achieve using the conventional PIC algorithms. In addition to summarizing the recent development and demonstrating practical implementations, several new results are also presented, including a structure-preserving geometric relativistic PIC algorithm, the proof of the correspondence between discrete gauge symmetry and discrete charge conservation law, and a reformulation of the explicit non-canonical symplectic algorithm for the discrete Poisson bracket using the variational approach. Numerical examples are given to verify the advantages of the structure- preserving geometric PIC algorithms in comparison with the conventional PIC methods.展开更多
The Pfaff-Birkhoff variational principle is discretized, and based on the discrete variational principle the discrete Birkhoffian equations are obtained. Taking the discrete equations as an algorithm, the correspondin...The Pfaff-Birkhoff variational principle is discretized, and based on the discrete variational principle the discrete Birkhoffian equations are obtained. Taking the discrete equations as an algorithm, the corresponding discrete flow is proved to be symplectic. That means the algorithm preserves the symplectic structure of Birkhofflan systems. Finally, simulation results of the given example indicate that structure-preserving algorithms have great advantage in stability and energy conserving.展开更多
基金supported by National Natural Science Foundation of China (Nos. 11975068 and 11925501)the National Key R&D Program of China (No. 2022YFE03090000)the Fundamental Research Funds for the Central Universities (No. DUT22ZD215)。
文摘The classical Pauli particle(CPP) serves as a slow manifold, substituting the conventional guiding center dynamics. Based on the CPP, we utilize the averaged vector field(AVF) method in the computations of drift orbits. Demonstrating significantly higher efficiency, this advanced method is capable of accomplishing the simulation in less than one-third of the time of directly computing the guiding center motion. In contrast to the CPP-based Boris algorithm, this approach inherits the advantages of the AVF method, yielding stable trajectories even achieved with a tenfold time step and reducing the energy error by two orders of magnitude. By comparing these two CPP algorithms with the traditional RK4 method, the numerical results indicate a remarkable performance in terms of both the computational efficiency and error elimination. Moreover, we verify the properties of slow manifold integrators and successfully observe the bounce on both sides of the limiting slow manifold with deliberately chosen perturbed initial conditions. To evaluate the practical value of the methods, we conduct simulations in non-axisymmetric perturbation magnetic fields as part of the experiments,demonstrating that our CPP-based AVF method can handle simulations under complex magnetic field configurations with high accuracy, which the CPP-based Boris algorithm lacks. Through numerical experiments, we demonstrate that the CPP can replace guiding center dynamics in using energy-preserving algorithms for computations, providing a new, efficient, as well as stable approach for applying structure-preserving algorithms in plasma simulations.
基金Project supported by the National Natural Science Foundation of China (Grant No 10572021)the Doctoral Programme Foundation of Institute of Higher Education of China (Grant No 20040007022)
文摘In this paper, the dissipative and the forced terms of the Duffing equation are considered as the perturbations of nonlinear Hamiltonian equations and the perturbational effect is indicated by parameter ε. Firstly, based on the gradient- Hamiltonian decomposition theory of vector fields, by using splitting methods, this paper constructs structure-preserving algorithms （SPAs） for the Duffing equation. Then, according to the Liouville formula, it proves that the Jacobian matrix determinants of the SPAs are equal to that of the exact flow of the Duffing equation. However, considering the explicit Runge Kutta methods, this paper finds that there is an error term of order p＋l for the Jacobian matrix determinants. The volume evolution law of a given region in phase space is discussed for different algorithms, respectively. As a result, the sum of Lyapunov exponents is exactly invariable for the SPAs proposed in this paper. Finally, through numerical experiments, relative norm errors and absolute energy errors of phase trajectories of the SPAs and the Heun method （a second-order Runge-Kutta method） are compared. Computational results illustrate that the SPAs are evidently better than the Heun method when e is small or equal to zero.
基金supported by National Natural Science Foundation of China (NSFC-11775219, 11775222, 11505186, 11575185 and 11575186)the National Key Research and Development Program (2016YFA0400600, 2016YFA0400601 and 2016YFA0400602)+3 种基金the ITER-China Program (2015GB111003, 2014GB124005)Chinese Scholar Council (201506340103)China Postdoctoral Science Foundation (2017LH002)the GeoA lgorithmic Plasma Simulator (GAPS) Project
文摘Recent development of structure-preserving geometric particle-in-cell （PIC） algorithms for Vlasov-Maxwell systems is summarized. With the arrival of 100 petaflop and exaflop computing power, it is now possible to carry out direct simulations of multi-scale plasma dynamics based on first-principles. However, standard algorithms currently adopted by the plasma physics community do not possess the long-term accuracy and fidelity required for these large-scale simulations. This is because conventional simulation algorithms are based on numerically solving the underpinning differential （or integro-differential） equations, and the algorithms used in general do not preserve the geometric and physical structures of the systems, such as the local energy-momentum conservation law, the symplectic structure, and the gauge symmetry. As a consequence, numerical errors accumulate coherently with time and long-term simulation results are not reliable. To overcome this difficulty and to harness the power of exascale computers, a new generation of structure-preserving geometric PIC algorithms have been developed. This new generation of algorithms utilizes modem mathematical techniques, such as discrete manifolds, interpolating differential forms, and non-canonical symplectic integrators, to ensure gauge symmetry, space-time symmetry and the conservation of charge, energy-momentum, and the symplectic structure. These highly desired properties are difficult to achieve using the conventional PIC algorithms. In addition to summarizing the recent development and demonstrating practical implementations, several new results are also presented, including a structure-preserving geometric relativistic PIC algorithm, the proof of the correspondence between discrete gauge symmetry and discrete charge conservation law, and a reformulation of the explicit non-canonical symplectic algorithm for the discrete Poisson bracket using the variational approach. Numerical examples are given to verify the advantages of the structure- preserving geometric PIC algorithms in comparison with the conventional PIC methods.
基金Supported by the National Natural Science Foundation of China (10932002,10972031)
文摘The Pfaff-Birkhoff variational principle is discretized, and based on the discrete variational principle the discrete Birkhoffian equations are obtained. Taking the discrete equations as an algorithm, the corresponding discrete flow is proved to be symplectic. That means the algorithm preserves the symplectic structure of Birkhofflan systems. Finally, simulation results of the given example indicate that structure-preserving algorithms have great advantage in stability and energy conserving.
