Article
Mathematics, Applied
Gengen Zhang, Chunmei Su
Summary: This paper introduces a highly accurate conservative method for solving the quantum Zakharov system, which is fourth-order accurate in space and second-order accurate in time according to detailed numerical analysis. The proposed scheme's conservation properties and high accuracy are confirmed through various numerical examples. Additionally, the compact scheme is used to study the convergence rate of the quantum Zakharov system to its limiting model in the semi-classical limit.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Yao Shi, Qiang Ma, Xiaohua Ding
Summary: This paper introduces a high-accuracy conservative difference scheme for solving the space fractional Zakharov system, with proven convergence rates. Numerical examples show the effectiveness of the scheme and validate theoretical results, highlighting the effects of fractional orders alpha and beta on solitary solution behaviors through intuitive images.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2021)
Article
Mathematics, Applied
Dongdong Hu, Huiling Jiang, Zhuangzhi Xu, Yushun Wang
Summary: This paper introduces a novel auxiliary variable approach and applies it to reformulate the fractional nonlinear Schrodinger equation and the coupled fractional nonlinear Klein-Gordon-Schrodinger equation. Extensive numerical experiments verify the correctness and efficiency of the proposed approach.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Jianqiang Xie, Muhammad Aamir Ali, Zhiyue Zhang
Summary: This paper focuses on the error estimation of a novel time second-order splitting conservative finite difference method for high-dimensional nonlinear fractional Schrodinger equation. The paper demonstrates the discrete preservation property and shows the accuracy of the method in terms of L2-norm. Numerical experiments are conducted to validate the accuracy and conservation property of the proposed method.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Jianqiang Xie, Quanxiang Wang, Zhiyue Zhang
Summary: Novel linearized implicit difference schemes with energy conservation property are constructed to simulate the propagation of fractional Klein-Gordon-Zakharov system. The convergence of the constructed algorithms is proven using discrete energy method. Numerical examples demonstrate the effectiveness of the suggested schemes.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Computer Science, Interdisciplinary Applications
Zhaopeng Hao, Zhongqiang Zhang, Rui Du
Summary: This study introduces a finite difference method for solving the fractional diffusion equation and analyzes its stability and convergence. It also presents a fast solver and provides numerical results to support the theoretical findings.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Interdisciplinary Applications
Yayun Fu, Qianqian Zheng, Yanmin Zhao, Zhuangzhi Xu
Summary: A family of high-order linearly implicit exponential integrators conservative schemes is proposed for solving the multi-dimensional nonlinear fractional Schrodinger equation. By reformulating and discretizing the equation, energy-preserving schemes with high accuracy are constructed to efficiently perform long-time simulations.
FRACTAL AND FRACTIONAL
(2022)
Article
Computer Science, Interdisciplinary Applications
Maohua Ran, Xiaoyi Zhou
Summary: This paper develops an efficient finite difference scheme for solving the time-fractional Cahn-Hilliard equations, and numerical experiments verify its stability, convergence, dynamics of the solution, and accuracy of the schemes. The solution of the time-fractional Cahn-Hilliard equation tends to an equilibrium state with the increase of time, consistent with the phase separation phenomenon.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2021)
Article
Mathematics, Applied
Changqing Ye, Junzhi Cui
Summary: In this paper, it is proved that Dziuk's fully discrete linearly implicit scheme has an optimal error estimate in the H-1 norm for curve shortening flow.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2021)
Article
Mathematics, Applied
Mingcong Xiao, Zhibo Wang, Yan Mo
Summary: In this paper, a fully implicit nonlinear difference scheme is proposed to solve the two-dimensional time fractional Burgers' equation with time delay based on the L1 discretization for the Caputo fractional derivative. The scheme has (2 - a)-th order accuracy in time and second-order accuracy in space, where a ? (0,1) is the fractional order. The existence of the numerical scheme is studied by the Browder fixed point theorem, and the unconditional stability and convergence of the scheme in L-2 norm are verified using the energy method and a fractional Gronwall inequality. A numerical example is also provided to illustrate the correctness of the theoretical analysis.
JOURNAL OF APPLIED MATHEMATICS AND COMPUTING
(2023)
Article
Mathematics, Applied
Yanyan Wang, Zhaopeng Hao, Rui Du
Summary: In this paper, a conservative three-layer linearized difference scheme for the two-dimensional nonlinear Schrodinger equation with fractional Laplacian is proposed. The scheme is proven to be uniquely solvable and it conserves mass and energy in the discrete sense. The scheme is also shown to be unconditionally convergent and stable under l(infinity)-norm, with a convergence order of O(tau(2) + h(2)).
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Pinxia Wu, Kejia Pan, Weiwei Ling, Qihong Wu
Summary: This work proposes a second-order finite difference (FD) scheme for the three-dimensional (3D) nonlinear Fitzhugh-Nagumo (FN) equation, with the nonlinear term treated using a semi-implicit technique. The existence and uniqueness of the difference scheme are proven, and the stability and convergence of the numerical solution are demonstrated. Additionally, an efficient extrapolation cascadic multigrid (EXCMG) method is employed to solve the large linear system arising from the FD discretization. Numerical results confirm the theoretical findings of the difference scheme and the efficiency of the EXCMG method, which can also be extended to solve other types of time-dependent nonlinear partial differential equations.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Anatoly A. Alikhanov, Chengming Huang
Summary: This paper focuses on constructing L2 type difference analog of the Caputo fractional derivative, studying its fundamental features, and using it to generate difference schemes with different orders in space and time for time fractional diffusion equations. The stability and convergence of the schemes are proven, and numerical computations support the obtained results.
APPLIED MATHEMATICS AND COMPUTATION
(2021)
Article
Mathematics, Applied
Hongling Hu, Xianlin Jin, Dongdong He, Kejia Pan, Qifeng Zhang
Summary: This paper introduces a linearized semi-implicit finite difference scheme for solving the two-dimensional space fractional nonlinear Schrodinger equation, which is characterized by mass and energy conservation, stability, and high accuracy. The optimal pointwise error estimate for the equation is rigorously established for the first time, along with a novel technique for handling the nonlinear term. The numerical results validate the theoretical findings.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
Leijie Qiao, Bo Tang
Summary: This paper presents an L1 implicit difference scheme based on non-uniform meshes for solving the time-fractional Burgers equation. The difficulty caused by the singularity of the exact solution at t = 0 can be overcome with non-uniform meshes. Through the energy method, the paper derives the unconditional stability and optimal convergence rate, with numerical experiments confirming the theoretical estimate.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Mathematics, Applied
Xuesong Wang, Yao Cheng
Summary: The local discontinuous Galerkin (LDG) method on a Shishkin mesh is investigated for a one-dimensional singularly perturbed reaction-diffusion problem. Improved pointwise error estimates are derived based on the discrete Green's function in the regular and layer regions. The convergence rates of the pointwise error for both the LDG approximation to the solution and its derivative are analyzed, showing optimal rates in different domains. Moreover, optimal pointwise error estimates are established when the regular component of the exact solution belongs to the finite element space. Numerical experiments are conducted to validate the theoretical findings.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Arash Goligerdian, Mahmood Khaksar-e Oshagh
Summary: This paper presents a computational method for simulating more accurate models for population growth with immigration, using integral equations with a delay parameter. The method utilizes Legendre wavelets within the Galerkin scheme as an orthonormal basis and employs the composite Gauss-Legendre quadrature rule for computing integrals. An error bound analysis demonstrates the convergence rate of the method, and various numerical examples are provided to validate the efficiency and accuracy of the technique as well as the theoretical error estimate.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
A. Sreelakshmi, V. P. Shyaman, Ashish Awasthi
Summary: This paper focuses on constructing a lucid and utilitarian approach to solve linear and non-linear two-dimensional partial differential equations. Through testing, it is found that the proposed method is highly applicable and accurate, showing excellent performance in terms of cost-cutting and time efficiency.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Shujiang Tang
Summary: This paper investigates the impact of the structure of local smoothness indicators on the computational performance of the WENO-Z scheme. A new class of two-parameter local smoothness indicators is proposed, which combines the classical WENO-JS and WENO-UD5 schemes and appends the coefficients of higher-order terms. A new WENO scheme, WENO-NSLI, is constructed using the global smoothness indicators of WENO-UD5. Numerical experiments show that the new scheme achieves optimal accuracy and has higher resolution compared to WENO-JS, WENO-Z, and WENO-UD5.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Xue-Feng Duan, Yong-Shen Zhang, Qing-Wen Wang
Summary: This paper addresses a class of constrained tensor least squares problems in image restoration and proposes the alternating direction multiplier method (ADMM) to solve them. The convergence analysis of this method is presented. Numerical experiments show the feasibility and effectiveness of the ADMM method for solving constrained tensor least squares problems, and simulation experiments on image restoration are also conducted.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Wanying Mao, Qifeng Zhang, Dinghua Xu, Yinghong Xu
Summary: In this paper, we derive, analyze, and extensively test fourth-order compact difference schemes for the Rosenau equations in one and two dimensions. These schemes are applied under spatial periodic boundary conditions using the double reduction order method and bilinear compact operator. Our results show that these schemes satisfy mass and energy conservation laws and have unique solvability, unconditional convergence, and stability. The convergence order is four in space and two in time under the D infinity-norm. Several numerical examples are provided to support the theoretical findings.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Jeremy Chouchoulis, Jochen Schutz
Summary: This work presents an approximate family of implicit multiderivative Runge-Kutta time integrators for stiff initial value problems and investigates two different methods for computing higher order derivatives. Numerical results demonstrate that adding separate formulas yields better performance in dealing with stiff problems.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Hui Yang, Shengfeng Zhu
Summary: In this paper, shape optimization in incompressible Stokes flows is investigated based on the penalty method for the divergence free constraint at continuous level. Shape sensitivity analysis is performed, and numerical algorithms are introduced. An iterative penalty method is used for solving the penalized state and possible adjoint numerically, and it is shown to be more efficient than the standard mixed finite element method in 2D. Asymptotic convergence analysis and error estimates for finite element discretizations of both state and adjoint are provided, and numerical results demonstrate the effectiveness of the optimization algorithms.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Reza MohammadiArani, Mehdi Dehghan, Mostafa Abbaszadeh
Summary: Lattice Boltzmann method is a powerful solver for fluid flow, but it is challenging to use it to solve other partial differential equations. This paper challenges the LBM to solve the two-dimensional DKS equation by finding a suitable local equilibrium distribution function and proposes a modification for implementing boundary conditions in complex geometries.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Arijit Das, Prakrati Kushwah, Jitraj Saha, Mehakpreet Singh
Summary: A new volume and number consistent finite volume scheme is introduced for the numerical solution of a collisional nonlinear breakage problem. The scheme achieves number consistency by introducing a single weight function in the flux formulation. The proposed scheme is efficient and robust, allowing easy coupling with computational fluid dynamics softwares.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
H. Ait el Bhira, M. Kzaz, F. Maach, J. Zerouaoui
Summary: We present an asymptotic method for efficiently computing second-order telegraph equations with high-frequency extrinsic oscillations. The method uses asymptotic expansions in inverse powers of the oscillatory parameter and derives coefficients through either recursion or solving non-oscillatory problems, leading to improved performance as the oscillation frequency increases.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Hanen Boujlida, Kaouther Ismail, Khaled Omrani
Summary: This study investigates a high-order accuracy finite difference scheme for solving the one-dimensional extended Fisher-Kolmogorov (EFK) equation. A new compact difference scheme is proposed and the a priori estimates and unique solvability are discussed using the discrete energy method. The unconditional stability and convergence of the difference solution are proved. Numerical experiments demonstrate the accuracy and efficiency of the proposed technique.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Alexander Zlotnik, Timofey Lomonosov
Summary: This paper studies a three-level explicit in time higher-order vector compact scheme for solving initial-boundary value problems for the n-dimensional wave equation and acoustic wave equation with variable speed of sound. By using additional sought functions to approximate second order non-mixed spatial derivatives of the solution, new stability bounds and error bounds of orders 4 and 3.5 are rigorously proved. Generalizations to nonuniform meshes in space and time are also discussed.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Fengli Yin, Yayun Fu
Summary: This paper develops an explicit energy-preserving scheme for solving the coupled nonlinear Schrodinger equation by combining the Lie-group method and GSAV approaches. The proposed scheme is efficient, accurate, and can preserve the modified energy of the system.
APPLIED NUMERICAL MATHEMATICS
(2024)