Article
Computer Science, Interdisciplinary Applications
Xavier Antoine, Jie Shen, Qinglin Tang
Summary: The paper proposes two linear implicit pseudo-spectral schemes for simulating the dynamics of general nonlinear Schrodinger/Gross-Pitaevskii equations, based on the SAV and LagM approaches. The SAV scheme preserves a modified total energy, while the LagM scheme can exactly preserve mass and original total energy. Each scheme has its own advantages in solving nonlinear algebraic systems and conserving errors.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
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
Xiujun Cheng, Hongyu Qin, Jiwei Zhang
Summary: This paper focuses on the construction and analysis of energy-conserving numerical schemes for generalized nonlinear space fractional Schrodinger equations with wave operator. An energy-conserving and linearly implicit scheme is presented, and numerical examples confirm the effectiveness of the proposed scheme in conserving system energy.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Ruize Yang, Yulong Xing
Summary: In this paper, a fully-discrete energy-conserving scheme for the nonlinear Dirac equation is proposed, which combines the SAV technique with DG discretization. The scheme conserves charge, energy exactly, and preserves multi-symplectic structure, demonstrating optimal convergence rates and energy-conserving property in numerical experiments.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Ahmed S. Hendy, J. E. Macias-Diaz
Summary: This study proposes a nonlinear conservative numerical method based on the SAV method to solve Hamiltonian space-fractional wave equations, with an associated discrete energy conservation property.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Mathematics, Applied
Xiaoxi Li, Jinming Wen, Dongfang Li
Summary: This paper introduces a fully discrete and structure-preserving scheme for the nonlinear fractional Schrodinger equations, showing that the scheme conserves mass and energy. This is in sharp contrast to the previous result where the fully discrete scheme only conserved mass for the original equations.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Mathematics, Applied
Alexandre Poulain, Katharina Schratz
Summary: This article carries out the convergence analysis of the scalar auxiliary variable method applied to the nonlinear Schrodinger equation. It presents weak and strong convergence results and provides error estimates and comparisons on energy conservation.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2022)
Article
Mathematics, Applied
Longzhao Qi, Yanren Hou
Summary: In this work, we design first-order and second-order time-stepping schemes for the modified phase field crystal model based on the scalar auxiliary variable method. The model is a nonlinear sixth-order damped wave equation that includes both elastic interactions and diffusive dynamics. Our schemes are linear and ensure unconditional energy stability with respect to pseudo energy. We also rigorously estimate the errors of the numerical schemes and present numerical tests to validate our theoretical results.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Junjie Wang
Summary: The paper presents high-order conservative schemes for the space fractional nonlinear Schrodinger equation, demonstrating their effectiveness through numerical experiments and proving the convergence of approximate solutions and the preservation of mass and energy conservation laws.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Mathematics, Applied
Shu Ma, Jilu Wang, Mingyan Zhang, Zhimin Zhang
Summary: This paper proposes a fully discrete finite element method with a Gauss collocation in time to solve the nonlinear Schrödinger equation. The method, based on the scalar auxiliary variable technique, preserves mass and energy conservations at the discrete level. Numerical results show the effectiveness and good convergence of the method.
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Yaping Li, Weidong Zhao, Wenju Zhao
Summary: In this paper, a scalar auxiliary variable approach combined with the discontinuous Galerkin method is proposed to handle the gradient-type nonlinear term in a nonlinear convection-diffusion equation. The proposed approach effectively incorporates spatial and temporal information to handle the nonlinear convection term and ensures system stability. The optimal accuracy is achieved with the discontinuous Galerkin method in space, and two different time discretization techniques are considered with first and second order accuracy. The proposed schemes are unconditionally stable, and optimal convergence rates are rigorously proved through error estimates. Numerical experiments confirm the convergence and demonstrate the robustness of the proposed approach in a benchmark problem with shock tendency.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
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
Mathematics, Applied
Hanzhang Hu, Yanping Chen
Summary: Two-grid algorithms based on two conservative and implicit finite element methods are studied for the two-dimensional nonlinear Schrodinger equation with wave operator. Both algorithms have three steps involving Newton iterations and error corrections. Optimal order L-p error estimations are conducted for error analysis of the algorithms without any time-step size conditions.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Engineering, Electrical & Electronic
Lanre Akinyemi, Mustafa Inc, Mostafa M. A. Khater, Hadi Rezazadeh
Summary: In this work, the exact traveling wave solutions of the (2 + 1)-dimensional Chiral nonlinear Schrodinger equation were studied using the generalized auxiliary equation method. The results showed that the aforementioned model has wide applications in quantum field theory, and the suggested technique provides various types of solutions.
OPTICAL AND QUANTUM ELECTRONICS
(2022)
Article
Materials Science, Multidisciplinary
Asif Khan, Amir Ali, Shabir Ahmad, Sayed Saifullah, Kamsing Nonlaopon, Ali Akgul
Summary: In this article, the behaviour of the time fractional nonlinear Schrodinger equation under two different operators are investigated. Numerical and analytical solutions are obtained using the modified double Laplace transform. The error analysis shows that the system depends primarily on time, with small errors observed for small time values. The efficiency of the proposed scheme is verified with examples and further analyzed graphically and numerically.
RESULTS IN PHYSICS
(2022)
Article
Mathematics, Applied
Junfeng Cao, Ke Chen, Huan Han
Summary: This paper proposes a two-stage image segmentation model based on structure tensor and fractional-order regularization. In the first stage, fractional-order regularization is used to approximate the Hausdorff measure of the MS model. The solution is found using the ADI scheme. In the second stage, thresholding is used for target segmentation. The proposed model demonstrates superior performance compared to state-of-the-art methods.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Dylan J. Oliver, Ian W. Turner, Elliot J. Carr
Summary: This paper discusses a projection-based framework for numerical computation of advection-diffusion-reaction (ADR) equations in heterogeneous media with multiple layers or complex geometric structures. By obtaining approximate solutions on a coarse grid and reconstructing solutions on a fine grid, the computational cost is significantly reduced while accurately approximating complex solutions.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Nathan V. Roberts, Sean T. Miller, Stephen D. Bond, Eric C. Cyr
Summary: In this study, the time-marching discontinuous Petrov-Galerkin (DPG) method is applied to the Vlasov equation for the first time, using backward Euler for a Vlasov-Poisson discretization. Adaptive mesh refinement is demonstrated on two problems: the two-stream instability problem and a cold diode problem.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Yizhi Sun, Zhilin Sun
Summary: This work investigates the convexity of a specific class of positive definite probability measures and demonstrates the preservation of convexity under multiplication and intertwining product. The study reveals that any integrable function on an interval with a polynomial expansion of fast absolute convergence can be decomposed into a pair of positive convex interval probabilities, simplifying the study of interval distributions and discontinuous probabilistic Galerkin schemes.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Bhagwan Singh, Komal Jangid, Santwana Mukhopadhyay
Summary: This paper examines the prediction of bending characteristics of nanoscale materials using the Moore-Gibson-Thompson thermoelasticity theory in conjunction with the nonlocal strain gradient theory. The study finds that the stiffness of the materials can be affected by nonlocal and length-scale parameters, and the aspect ratios of the beam structure play a significant role in bending simulations.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Guoliang Wang, Bo Zheng, Yueqiang Shang
Summary: This paper presents and analyzes a parallel finite element post-processing algorithm for the simulation of Stokes equations with a nonlinear damping term, which integrates the algorithmic advantages of the two-level approach, the partition of unity method, and the post-processing technique. The algorithm generates a global continuous approximate solution using the partition of unity method and improves the smoothness of the solution by adding an extra coarse grid correction step. It has good parallel performance and is validated through theoretical error estimates and numerical test examples.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Hao Xu, Zeng-Qi Wang
Summary: Fluid flow control problems are crucial in industrial applications, and solving the optimal control of Navier-Stokes equations is challenging. By using Oseen's approximation and matrix splitting preconditioners, we can efficiently solve the linear systems and improve convergence.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Zhengya Yang, Xuejuan Chen, Yanping Chen, Jing Wang
Summary: This paper focuses on the high-order stable numerical solutions of the time-space fractional diffusion equation. The Fourier spectral method is used for spatial discretization and the Spectral Deferred Correction (SDC) method is used for numerical solutions in time. As a result, a high-precision numerical discretization scheme for solving the fractional diffusion equation is obtained, and the convergence and stability of the scheme are proved. Several numerical examples are presented to demonstrate the effectiveness and feasibility of the proposed numerical scheme.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)