Article
Mathematics, Applied
Dongdong Hu, Wenjun Cai, Yushun Wang
Summary: In this paper, two linearly implicit energy preserving schemes with constant coefficient matrix for multi-dimensional fractional nonlinear Schrodinger equations are proposed. By introducing an exponential auxiliary variable and utilizing the Lawson transformation, equivalent systems with mass and energy conservation laws are formulated. The numerical schemes demonstrate high efficiency in energy preservation for long-time computations with second-order accuracy in time and spectral accuracy in space.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Computer Science, Interdisciplinary Applications
Longbin Wu, Qiang Ma, Xiaohua Ding
Summary: This paper presents an energy-preserving scheme for the nonlinear fractional Klein-Gordon Schrodinger equation using the scalar auxiliary variable approach. By introducing a scalar variable, the system is transformed into a new equivalent system, and a linear implicit energy-preserving scheme is obtained by applying the extrapolated Crank-Nicolson method in the temporal direction and Fourier pseudospectral method in the spatial direction.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2021)
Review
Mathematics, Applied
Yayun Fu, Yanhua Shi, Yanmin Zhao
Summary: This paper aims to construct a class of high-order explicit conservative schemes for the space fractional nonlinear Schrodinger equation by combining the invariant energy quadratization method and Runge-Kutta method. Numerical experiments demonstrate the conservative properties, convergence orders, and long time stability of the proposed schemes, which provide a promising approach for solving the equation.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2022)
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
Computer Science, Interdisciplinary Applications
Lili Ju, Xiao Li, Zhonghua Qiao, Jiang Yang
Summary: This paper investigates high-order MBP-preserving time integration schemes using the integrating factor Runge-Kutta (IFRK) method and shows that the classic four-stage, fourth-order IFRK scheme is MBP preserving for some typical semilinear systems, although not strong stability preserving. The efficiency and convergence of these numerical schemes are theoretically proved and numerically verified, with simulations conducted on 2D and 3D long-time evolutional behaviors, including a model that is not a typical gradient flow as the Allen-Cahn type of equations.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Computer Science, Interdisciplinary Applications
Caixia Nan, Huailing Song
Summary: This paper applies the explicit integrating factor Runge-Kutta methods (eIFRK(+)) to the nonlocal Allen-Cahn (NAC) equation and proposes new eIFRK(+) schemes. The method is shown to preserve crucial physical properties of the NAC model under a large time-step constraint, and its effectiveness is validated through theoretical analysis and numerical experiments.
JOURNAL OF COMPUTATIONAL PHYSICS
(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
Mathematics, Interdisciplinary Applications
Jin Cui, Yayun Fu
Summary: In this paper, we propose a novel class of high-order, linearly implicit, and energy-preserving numerical schemes for solving nonlinear dispersive equations. By applying the energy quadratization technique, the original system is transformed into an equivalent system with quadratization energy. The reformulated system is then discretized in time using the prediction-correction strategy and the Partitioned Runge-Kutta method. The resulting semi-discrete system is high-order, linearly implicit, and exactly preserves the quadratic energy of the reformulated system. The efficiency and accuracy of the proposed schemes are demonstrated using the Camassa-Holm equation as a benchmark.
NETWORKS AND HETEROGENEOUS MEDIA
(2023)
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
Ben S. Southworth, Oliver Krzysik, Will Pazner, Hans De Sterck
Summary: This paper introduces a theoretical and algorithmic preconditioning framework for solving the systems of equations that arise from fully implicit Runge-Kutta methods applied to linear numerical PDEs. The preconditioned operator is proven to have a condition number bounded by a small constant, independent of the spatial mesh and time-step size, and with weak dependence on number of stages/polynomial order.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(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
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
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)
Article
Mathematics, Applied
Hong Zhang, Jingye Yan, Xu Qian, Songhe Song
Summary: This paper addresses the open problem of whether high order temporal integrators can preserve the maximum principle of the Allen-Cahn equation by designing and analyzing a class of up to fourth order maximum principle preserving integrators. The proposed method, which combines second order finite difference discretization and Runge-Kutta integration, is proven to converge with order O(tau(p) + h(2)) in the discrete L-infinity norm. Various experiments in 1D, 2D, and 3D problems demonstrate the high order convergence and maximum principle preserving capability of the algorithms over a long time.
APPLIED NUMERICAL MATHEMATICS
(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)