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)
Article
Mathematics, Applied
Yuezheng Gong, Qi Hong, Chunwu Wang, Yushun Wang
Summary: In this paper, a quadratic auxiliary variable (QAV) technique is used to develop energy-preserving algorithms for the Camassa-Holm equation. The technique transforms the original equation and discretizes it to obtain a class of fully discrete schemes. The proposed methods are proven to be energy-preserving, and numerical experiments confirm their accuracy, conservative property, and efficiency.
ADVANCES IN APPLIED MATHEMATICS AND MECHANICS
(2022)
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
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
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
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
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
Computer Science, Interdisciplinary Applications
E. M. J. Komen, J. A. Hopman, E. M. A. Frederix, F. X. Trias, R. W. C. P. Verstappen
Summary: A new conservative symmetry-preserving second-order time-accurate PISO-based method for solving the incompressible Navier-Stokes equations on unstructured collocated grids is presented and compared with an explicit method. The new method shows advantages in energy conservation and numerical stability, making it potentially valuable for high-fidelity simulations in complex geometries. Implemented in the widely used OpenFOAM, these methods are expected to benefit the CFD community by paving the way for truly energy-conservative high-fidelity simulations.
COMPUTERS & FLUIDS
(2021)
Article
Mathematics, Applied
S. Blanes, F. Casas, A. Escorihuela-Tomas
Summary: Different families of Runge-Kutta-Nystrom (RKN) symplectic splitting methods of order 8 are presented and tested for second-order systems of ordinary differential equations. They demonstrate better efficiency than state-of-the-art symmetric compositions of 2nd-order symmetric schemes and RKN splitting methods of orders 4 and 6, particularly for medium to high accuracy. In some specific examples, they are even more efficient than extrapolation methods for high accuracies and integrations over relatively short time intervals.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Engineering, Electrical & Electronic
Nauman Raza, Zara Hassan, Aly Seadawy
Summary: The Collective Variable (CV) approach is introduced to analyze a significant form of Schrodinger equation with variable coefficients and higher order effects. Numerical simulations using the Runge-Kutta method of order four are implemented to explore pulse parameters, showing fluctuations in pulse variables and periodicity in chirp, width, amplitude, phase, and frequency of soliton. Different values of pulse parameters demonstrate variations in collective variables of solitons.
OPTICAL AND QUANTUM ELECTRONICS
(2021)
Article
Mathematics, Applied
Yonghui Bo, Wenjun Cai, Yushun Wang
Summary: Symplectic schemes applied to Hamiltonian systems have prominent advantages for preserving qualitative properties of the flow, with the symplectic Euler, implicit midpoint, and Stormer-Verlet methods being the simplest and widely used. This paper introduces a simple symplectic scheme with three free parameters, along with a second-order symplectic scheme with two free parameters and a symmetric symplectic scheme with a free parameter. Adjusting the parameter at each time step allows for a second-order energy and quadratic invariants preserving method.
APPLIED MATHEMATICS LETTERS
(2021)
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
Qi Hong, Yuezheng Gong, Jia Zhao, Qi Wang
Summary: Researchers developed fully discrete, structure-preserving numerical algorithms of arbitrarily high order for the Allen-Cahn model with a nonlocal constraint, achieving the necessary numerical accuracy and efficiency through energy quadratization methodology. The reformulation of the model into an equivalent one with a quadratic free energy allowed for the preservation of volume conservation and energy dissipative property, while employing distinct temporal discretization methods for fully discrete schemes of arbitrarily higher order.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Mathematics, Applied
T. Dzanic, W. Trojak, F. D. Witherden
Summary: In this work, a modified explicit Runge-Kutta temporal integration scheme is proposed to guarantee the preservation of locally-defined quasiconvex set of bounds for the solution. The schemes use a bijective mapping to enforce bounds between the admissible set of solutions and the real domain. It is shown that these schemes can recover a wide range of methods, including positivity preserving, discrete maximum principle satisfying, entropy dissipative, and invariant domain preserving schemes. The additional computational cost is the evaluation of two nonlinear mappings which generally have closed-form solutions. The approach is demonstrated in numerical experiments using a pseudospectral spatial discretization without explicit shock capturing schemes for nonlinear hyperbolic problems with discontinuities.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Jun Yang, Nianyu Yi, Hong Zhang
Summary: Based on the mass-lumping finite element space discretization, this paper introduces a class of temporal up to the fourth-order unconditionally structure-preserving schemes for the Allen-Cahn equation and its conservative forms, by incorporating the integrating factor Runge-Kutta method and stabilization technique. The proposed methods are linear, do not require any post-processing or limiters, and unconditionally preserve the maximum principle and mass conservation law. Numerical experiments verify the high-order temporal accuracy and the ability to preserve the maximum principle, mass conservation, and energy stability over long periods. Additionally, numerical simulation demonstrates the good performance of the proposed schemes in terms of structure-preserving with high-order finite element method.
APPLIED NUMERICAL MATHEMATICS
(2023)