“…Remark 2.2 It is noted that this kind of method belongs to exponential integrators, which have been widely developed and researched for solving highly oscillatory systems (see, e.g. [19,20,21,24,30,32]).…”
In this paper, exponential energy-preserving methods are formulated and analysed for solving charged-particle dynamics in a strong and constant magnetic field. The resulting method can exactly preserve the energy of the dynamics. Moreover, it is shown that the magnetic moment of the considered system is nearly conserved over a long time along this exponential energy-preserving method, which is proved by using modulated Fourier expansions. Other properties of the method including symmetry and convergence are also studied. An illustrated numerical experiment is carried out to demonstrate the long-time behaviour of the method.
“…Remark 2.2 It is noted that this kind of method belongs to exponential integrators, which have been widely developed and researched for solving highly oscillatory systems (see, e.g. [19,20,21,24,30,32]).…”
In this paper, exponential energy-preserving methods are formulated and analysed for solving charged-particle dynamics in a strong and constant magnetic field. The resulting method can exactly preserve the energy of the dynamics. Moreover, it is shown that the magnetic moment of the considered system is nearly conserved over a long time along this exponential energy-preserving method, which is proved by using modulated Fourier expansions. Other properties of the method including symmetry and convergence are also studied. An illustrated numerical experiment is carried out to demonstrate the long-time behaviour of the method.
In some cases, high-order methods are known to provide greater accuracy with larger step-sizes than lower order methods. Hence, in this paper, we present a Block Hybrid Method (BHM) of order 11 for directly solving systems of general second-order initial value problems (IVPs), including Hamiltonian systems and partial differential equations (PDEs), which arise in multiple areas of science and engineering. The BHM is formulated from a continuous scheme based on a hybrid method of a linear multistep type with several off-grid points and then implemented in a block-by-block manner. The properties of the BHM are discussed and the performance of the method is demonstrated on some numerical examples. In particular, the superiority of the BHM over the Generalized Adams Method (GAM) of order 11 is established numerically.
“…This paper is devoted to analyze an efficient triangular splitting implementation of a special kind of RKN methods. We consider here the following k ‐stage RKN‐type Fourier collocation methods (firstly presented in Wang et al), when applied to : where h is the stepsize, r is an integer with the requirement 2≤ r ≤ k , are the shifted Legendre polynomials defined in [0,1], ( c l , b l ) with l =1,2,…, k are the node points and the quadrature weights of a quadrature formula, respectively. The method is the subclass of k ‐stage RKN method with the following Butcher tableau: …”
Section: Introductionmentioning
confidence: 99%
“…Such problems often arise in many fields of applied sciences such as applied mathematics, celestial mechanics, physics, quantum chemistry, and electronics. Many effective numerical methods have been developed and studied for solving the second-order system (1) such as Runge-Kutta-Nyström (RKN) methods and multistep methods (see, eg, previous studies [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] ). It is well known that an s-stage fully implicit RKN method involves an sd-dimensional nonlinear system.…”
Section: Introductionmentioning
confidence: 99%
“…This paper is devoted to analyze an efficient triangular splitting implementation of a special kind of RKN methods. We consider here the following k-stage RKN-type Fourier collocation methods (firstly presented in Wang et al 11 ), when applied to (1):…”
A triangular splitting implementation of Runge–Kutta–Nyström–type Fourier collocation methods is presented and analyzed in this paper. The proposed implementation relies on a reformulation of the method and on the Crout factorization of a corresponding matrix associated with the method. The excellent behavior of the splitting implementation is confirmed by its performance on a few numerical tests.
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