Efficient implementation of RKN-type Fourier collocation methods for second-order differential equations

Wang, Bin; Meng, Fanwei; Fang, Yonglei

doi:10.1016/j.apnum.2017.04.008

Cited by 53 publications

(35 citation statements)

References 24 publications

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“…To show the efficiency and robustness of the triangular splitting implementation, the iterations we select for comparison are: TRKNTCM1: the above method using Algorithm 3.4 with 1 inner iteration; TRKNTCM2: the above method using Algorithm 3.4 with 2 inner iterations; TRKNTCM3: the above method using Algorithm 3.4 with 3 inner iterations; FPRKNTCM: the above method using fixed‐point iteration; SNRKNTCM: the above method using simplified‐Newton iteration; ERKNTCM: the above method using the efficient algorithm given by Table 3 in Wang et al …”

Section: Numerical Testsmentioning

confidence: 99%

Triangular splitting implementation of RKN‐type Fourier collocation methods for second‐order differential equations

Wang

2018

Math Methods in App Sciences

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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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Section: Numerical Testsmentioning

confidence: 99%

Triangular splitting implementation of RKN‐type Fourier collocation methods for second‐order differential equations

Wang

2018

Math Methods in App Sciences

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“…Consequently, all possible Casimirs and the Hamiltonian H are conserved quantities for (50). In the sequel, we show that EQUIP methods can be conveniently used for solving such problems.…”

Section: Poisson Problemsmentioning

confidence: 99%

“…(28,48,50) 3.24 1800 11.1 2.25 × 10 −10 4.44 × 10 −16 (28,47,49) 3.49 1900 10.5 2.20 × 10 −10 4.44 × 10 −16 (27,45,47) 3.59 2000 10.0 2.89 × 10 −10 3.33 × 10 −16 (26,44,46) 3.46…”

Section: Highly Oscillatory Problemsmentioning

confidence: 99%

“…C is called a Casimir function for (50). Consequently, all possible Casimirs and the Hamiltonian H are conserved quantities for (50).…”

Section: Poisson Problemsmentioning

confidence: 99%

“…The latter code is also available on the Test Set for IVP Solvers [48] (see also [49]), and turns out to be among the most reliable and efficient codes currently available for solving stiff ODE-IVPs and linearly implicit DAEs. It is worth mentioning that, more recently, the blended implementation of RKN-type methods has been also considered [50].…”

Section: Blended Implementation Of Hbvmsmentioning

confidence: 99%

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Abstract:In recent years, the numerical solution of differential problems, possessing constants of motion, has been attacked by imposing the vanishing of a corresponding line integral. The resulting methods have been, therefore, collectively named (discrete) line integral methods, where it is taken into account that a suitable numerical quadrature is used. The methods, at first devised for the numerical solution of Hamiltonian problems, have been later generalized along several directions and, actually, the research is still very active. In this paper we collect the main facts about line integral methods, also sketching various research trends, and provide a comprehensive set of references.

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Exponential collocation methods based on continuous finite element approximations for efficiently solving the cubic Schrödinger equation

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2020

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In this paper we derive and analyze new exponential collocation methods to efficiently solve the cubic Schrödinger Cauchy problem on a d-dimensional torus. The novel methods are formulated based on continuous time finite element approximations in a generalized function space. Energy preservation is a key feature of the cubic Schrödinger equation. It is proved that the novel methods can be of arbitrarily high order which exactly or approximately preserve the continuous energy of the original continuous system. The existence and uniqueness, regularity, and convergence of the new methods are studied in detail. Two practical exponential collocation methods are constructed, and three illustrative numerical experiments are included. The numerical results show the remarkable accuracy and efficiency of the new methods in comparison with existing numerical methods in the literature.

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Efficient implementation of RKN-type Fourier collocation methods for second-order differential equations

Cited by 53 publications

References 24 publications

Triangular splitting implementation of RKN‐type Fourier collocation methods for second‐order differential equations

Triangular splitting implementation of RKN‐type Fourier collocation methods for second‐order differential equations

Line Integral Solution of Differential Problems

Exponential collocation methods based on continuous finite element approximations for efficiently solving the cubic Schrödinger equation

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