In this paper, a novel class of exponential Fourier collocation methods （EFCMs） is presented for solving systems of first-order ordinary differential equations. These so-called exponential Fourier collocation method...In this paper, a novel class of exponential Fourier collocation methods （EFCMs） is presented for solving systems of first-order ordinary differential equations. These so-called exponential Fourier collocation methods are based on the variation-of-constants formula, incorporating a local Fourier expansion of the underlying problem with collocation meth- ods. We discuss in detail the connections of EFCMs with trigonometric Fourier colloca- tion methods （TFCMs）, the well-known Hamiltonian Boundary Value Methods （HBVMs）, Gauss methods and Radau IIA methods. It turns out that the novel EFCMs are an es- sential extension of these existing methods. We also analyse the accuracy in preserving the quadratic invariants and the Hamiltonian energy when the underlying system is a Hamiltonian system. Other properties of EFCMs including the order of approximations and the convergence of fixed-point iterations are investigated as well. The analysis given in this paper proves further that EFCMs can achieve arbitrarily high order in a routine manner which allows us to construct higher-order methods for solving systems of first- order ordinary differential equations conveniently. We also derive a practical fourth-order EFCM denoted by EFCM（2,2） as an illustrative example. The numerical experiments using EFCM（2,2） are implemented in comparison with an existing fourth-order HBVM, an energy-preserving collocation method and a fourth-order exponential integrator in the literature. The numerical results demonstrate the remarkable efficiency and robustness of the novel EFCM（2,2）.展开更多
证明了n阶齐次线性微分方程(dnx)/(dtn)+a1(t)(dn-1x)/(dtn-1)+…+an-1(t)dx/dt+an(t)x=0的Liouville公式W′(t)=W(t0)e-∫tt0a1(s)ds是一阶齐次线性微分方程组x′=A(t)x所对应的Liouville公式W′(t)=W(t0)e-integral from a=1 to t sum...证明了n阶齐次线性微分方程(dnx)/(dtn)+a1(t)(dn-1x)/(dtn-1)+…+an-1(t)dx/dt+an(t)x=0的Liouville公式W′(t)=W(t0)e-∫tt0a1(s)ds是一阶齐次线性微分方程组x′=A(t)x所对应的Liouville公式W′(t)=W(t0)e-integral from a=1 to t sum from i=1 to n aii(s)ds的特殊情形。展开更多
文摘In this paper, a novel class of exponential Fourier collocation methods （EFCMs） is presented for solving systems of first-order ordinary differential equations. These so-called exponential Fourier collocation methods are based on the variation-of-constants formula, incorporating a local Fourier expansion of the underlying problem with collocation meth- ods. We discuss in detail the connections of EFCMs with trigonometric Fourier colloca- tion methods （TFCMs）, the well-known Hamiltonian Boundary Value Methods （HBVMs）, Gauss methods and Radau IIA methods. It turns out that the novel EFCMs are an es- sential extension of these existing methods. We also analyse the accuracy in preserving the quadratic invariants and the Hamiltonian energy when the underlying system is a Hamiltonian system. Other properties of EFCMs including the order of approximations and the convergence of fixed-point iterations are investigated as well. The analysis given in this paper proves further that EFCMs can achieve arbitrarily high order in a routine manner which allows us to construct higher-order methods for solving systems of first- order ordinary differential equations conveniently. We also derive a practical fourth-order EFCM denoted by EFCM（2,2） as an illustrative example. The numerical experiments using EFCM（2,2） are implemented in comparison with an existing fourth-order HBVM, an energy-preserving collocation method and a fourth-order exponential integrator in the literature. The numerical results demonstrate the remarkable efficiency and robustness of the novel EFCM（2,2）.
文摘证明了n阶齐次线性微分方程(dnx)/(dtn)+a1(t)(dn-1x)/(dtn-1)+…+an-1(t)dx/dt+an(t)x=0的Liouville公式W′(t)=W(t0)e-∫tt0a1(s)ds是一阶齐次线性微分方程组x′=A(t)x所对应的Liouville公式W′(t)=W(t0)e-integral from a=1 to t sum from i=1 to n aii(s)ds的特殊情形。
