Article
Mathematics, Applied
Gengen Zhang, Chaolong Jiang, Hao Huang
Summary: In this paper, a novel class of high-order energy-preserving schemes for solving the Zakharov-Rubenchik equations is proposed. The schemes introduce a quadratic auxiliary variable to transform the Hamiltonian energy and reformulate the original system into an equivalent system satisfying multiple invariants. The schemes achieve high-order accuracy in time and conserve the mass, Hamiltonian energy, and two linear invariants.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Review
Mathematics, Applied
Yayun Fu, Xuelong Gu, Yushun Wang, Wenjun Cai
Summary: We present a class of arbitrarily high-order conservative schemes for the Klein-Gordon Schrodinger equations, which combine the symplectic Runge-Kutta method with the quadratic auxiliary variable approach and can effectively preserve the conservation of energy and mass.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Xin Li, Yuezheng Gong, Luming Zhang
Summary: This paper develops two classes of linear high-order conservative numerical schemes for the nonlinear Schrodinger equation with wave operator. By utilizing the method of order reduction in time and scalar auxiliary variable technique, the original model is transformed into an equivalent system with modified energy as a quadratic form. Linear high-order energy-preserving schemes are constructed using extrapolation strategy and symplectic Runge-Kutta method in time, providing a paradigm for developing structure-preserving algorithms of arbitrarily high order.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Gengen Zhang, Chaolong Jiang
Summary: This paper presents a new methodology for developing arbitrary high-order structure-preserving methods to solve the quantum Zakharov system. The method reformulates the original Hamiltonian energy into a quadratic form by introducing a new quadratic auxiliary variable, and then rewrites the original system into a new equivalent system based on the energy variational principle. The proposed method achieves arbitrary high-order accuracy in time in a periodic domain and exactly preserves the discrete mass and original Hamiltonian energy.
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Yayun Fu, Dongdong Hu, Gengen Zhang
Summary: This paper proposes a family of high-order conservative schemes based on the exponential integrators technique and the symplectic Runge-Kutta method for solving the nonlinear Gross-Pitaevskii equation. Numerical examples are provided to confirm the accuracy and conservation of the developed schemes.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Yue Chen, Yuezheng Gong, Qi Hong, Chunwu Wang
Summary: In this paper, a quadratic auxiliary variable approach is proposed to develop energy-preserving Runge-Kutta methods for the Korteweg-de Vries equation. The approach reformulates the original model into an equivalent system and employs symplectic Runge-Kutta methods to obtain a new kind of time semi-discrete schemes. The proposed methods effectively preserve the energy conservation law and achieve efficient calculation with a iterative technique.
NUMERICAL MATHEMATICS-THEORY METHODS AND APPLICATIONS
(2022)
Article
Mathematics, Applied
Solve Eidnes, Lu Li, Shun Sato
Summary: The study investigates and compares Kahan's method and a two-step generalisation of the discrete gradient method, applied to the Korteweg-de Vries equation and the Camassa-Holm equation. Numerical results are presented and analysed in this investigation.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Physics, Fluids & Plasmas
Zhaoli Guo, Jiequan Li, Kun Xu
Summary: The kinetic theory serves as a foundation for the development of multiscale methods for gas flows. It is challenging for kinetic schemes to accurately capture the hydrodynamic behaviors of the system at the continuum regime without enforcing kinetic scale resolution. The concept of unified preserving (UP) is introduced to assess the asymptotic orders of a kinetic scheme and its dependence on spatial and temporal accuracy, as well as the interconnections among three scales: kinetic scale, numerical scale, and hydrodynamic scale.
Article
Mathematics, Applied
Chaolong Jiang, Xu Qian, Songhe Song, Chenxuan Zheng
Summary: Arbitrary high-order numerical schemes conserving the momentum and energy of the generalized Rosenau-type equation are studied in this paper. The momentum-preserving schemes are derived within the symplectic Runge-Kutta method coupled with the standard Fourier pseudo-spectral method. By combining the quadratic auxiliary variable approach, symplectic Runge-Kutta method, and standard Fourier pseudo-spectral method, a class of high-order mass-and energy-preserving schemes for the Rosenau equation is introduced. Various numerical tests demonstrate the performance of the proposed schemes.
EAST ASIAN JOURNAL ON APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Cristina Anton
Summary: This article gives conditions for stochastic Runge-Kutta methods to nearly preserve quadratic invariants and discusses the corresponding simplified order conditions. For stochastic Hamiltonian systems, a systematic approach is proposed to construct explicit stochastic Runge-Kutta pseudo-symplectic schemes based on colored trees and B-series. Pseudo-symplectic stochastic Runge-Kutta methods with strong convergence order are constructed, and the long-term performance of the proposed schemes is numerically illustrated.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Brian C. Vermeire, Siavash Hedayati Nasab
Summary: This paper introduces a family of accelerated implicit-explicit (AIMEX) schemes for solving stiff systems of equations. AIMEX schemes can significantly improve stability and allowable time step sizes.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Davide Torlo, Philipp Oeffner, Hendrik Ranocha
Summary: This article discusses the methods to analyze the performance and robustness of Patankar-type schemes, and demonstrates their problematic behavior on both linear and nonlinear stiff problems.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Brian C. Vermeire
Summary: Recently, Paired Explicit Runge-Kutta (P-ERK) schemes were introduced to accelerate the solution of locally-stiff systems of equations by significantly reducing computational cost. In this work, a framework for incorporating an embedded pair within the original P-ERK formulation is presented. Numerical results demonstrate that adaptive time stepping using embedded P-ERK schemes yields excellent agreement with reference data while being up to seven times less computationally expensive than classical embedded pairs for all cases.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics, Applied
Hong Zhang, Jingye Yan, Xu Qian, Xiaowei Chen, Songhe Song
Summary: This study focuses on the solution methods for improved conservative Allen-Cahn equations, proposing new structure-preserving schemes and isIFRK schemes. Numerical experiments confirm the advantages of isIFRK schemes, including high-order accuracy, mass conservation, and unconditional preservation of the maximum principle.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Shun Sato, Yuto Miyatake, John C. Butcher
Summary: In this paper, linearly implicit and arbitrary high-order conservative numerical schemes for ordinary differential equations with a quadratic invariant are proposed. Quadratic invariants are important objects appearing in many physical examples and computationally efficient conservative schemes. The authors construct such schemes based on canonical Runge-Kutta methods and prove some properties involving accuracy.
APPLIED NUMERICAL MATHEMATICS
(2023)