Superconvergence analysis of symmetric Gauss-type exponential collocation integrators for solving the multidimensional nonlinear first-order partial differential equations

The main objective of this research is to develop and analyze high-order symmetric Gauss-type exponential collocation time-stepping methods for solving systems of nonlinear first-order partial differential equations (PDEs). Initially, the nonlinear PDEs are reformulated as an abstract Hamiltonian or...

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Published in:	Journal of mathematical chemistry Vol. 63; no. 10; pp. 2023 - 2050
Main Authors:	Wu, Xiumei, Fang, Yonglei, Liu, Changying, Song, Yuanling
Format:	Journal Article
Language:	English
Published:	Cham Springer International Publishing 01.11.2025 Springer Nature B.V
Subjects:	Accuracy Boundary conditions Chemistry Chemistry and Materials Science Collocation Collocation methods Eigenvectors Function space Hamiltonian functions Hilbert space Integrators Math. Applications in Chemistry Nonlinear differential equations Nonlinear systems Ordinary differential equations Original Paper Parabolic differential equations Partial differential equations Physical Chemistry Symmetry Theoretical and Computational Chemistry 65L05 65L06 34C14 Nonlinear first-order PDEs 65L70 Variation-of-constants formula Exponential integrator 65L20 Superconvergence Abstract ordinary differential equation Symmetry
ISSN:	0259-9791, 1572-8897
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Abstract	The main objective of this research is to develop and analyze high-order symmetric Gauss-type exponential collocation time-stepping methods for solving systems of nonlinear first-order partial differential equations (PDEs). Initially, the nonlinear PDEs are reformulated as an abstract Hamiltonian ordinary differential equation (ODE) system in an appropriate infinite-dimensional function space. Subsequently, the Gauss-type exponential collocation time integrators are derived. The symmetry, local error bounds and nonlinear stability of the proposed time integrators are rigorously analysed in details. Furthermore, the rigourous convergence analysis demonstrates that Gauss-type exponential collocation time integrators can achieve superconvergence. Numerical experiments verify our theoretical analysis results, and demonstrate the remarkable superiority in comparison with the traditional temporal integration methods.
AbstractList	The main objective of this research is to develop and analyze high-order symmetric Gauss-type exponential collocation time-stepping methods for solving systems of nonlinear first-order partial differential equations (PDEs). Initially, the nonlinear PDEs are reformulated as an abstract Hamiltonian ordinary differential equation (ODE) system in an appropriate infinite-dimensional function space. Subsequently, the Gauss-type exponential collocation time integrators are derived. The symmetry, local error bounds and nonlinear stability of the proposed time integrators are rigorously analysed in details. Furthermore, the rigourous convergence analysis demonstrates that Gauss-type exponential collocation time integrators can achieve superconvergence. Numerical experiments verify our theoretical analysis results, and demonstrate the remarkable superiority in comparison with the traditional temporal integration methods.
Author	Fang, Yonglei Liu, Changying Wu, Xiumei Song, Yuanling
Author_xml	– sequence: 1 givenname: Xiumei surname: Wu fullname: Wu, Xiumei email: hzxywu06@163.com organization: School of Mathematics and Statistics, Heze University – sequence: 2 givenname: Yonglei surname: Fang fullname: Fang, Yonglei organization: School of Mathematics and Statistics, Zaozhuang University – sequence: 3 givenname: Changying surname: Liu fullname: Liu, Changying organization: School of Mathematics and Statistics, Heze University – sequence: 4 givenname: Yuanling surname: Song fullname: Song, Yuanling organization: Jiangsu Province Gaoyou No.1 Middle School
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Cites_doi	10.1017/CBO9780511618352 10.1137/S0036142995280572 10.1016/j.jcp.2017.03.038 10.1016/j.cam.2019.04.015 10.1017/S0962492910000048 10.1016/j.jcp.2017.10.057 10.1145/1206040.1206044 10.1016/j.camwa.2022.10.023 10.1007/BFb0067462 10.1016/j.apnum.2004.08.005 10.1016/0001-6160(79)90196-2 10.1007/s11075-024-01940-7 10.1137/04061101X 10.1137/0709060 10.1016/j.jcp.2017.03.018 10.1137/040611434 10.1016/j.aml.2020.106265 10.1007/978-3-540-71041-7 10.1007/BF01396634 10.1016/j.cam.2025.116586 10.1137/0704033 10.1137/15M1023257
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SubjectTerms	Accuracy Boundary conditions Chemistry Chemistry and Materials Science Collocation Collocation methods Eigenvectors Function space Hamiltonian functions Hilbert space Integrators Math. Applications in Chemistry Nonlinear differential equations Nonlinear systems Ordinary differential equations Original Paper Parabolic differential equations Partial differential equations Physical Chemistry Symmetry Theoretical and Computational Chemistry
Title	Superconvergence analysis of symmetric Gauss-type exponential collocation integrators for solving the multidimensional nonlinear first-order partial differential equations
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