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
Yongyong Cai, Jinxue Fu, Jianfeng Liu, Tingchun Wang
Summary: In this paper, a new fourth-order compact finite difference scheme is proposed for solving the quantum Zakharov system. The scheme preserves the mass and energy conservation properties of the system, while also preserving the energy at each time step. By analyzing the equivalent form of the difference scheme using the energy method, rigorous optimal error estimates are established in the energy norm. The scheme demonstrates fourth-order accuracy in space and second-order accuracy in time. Numerical examples validate the theoretical results and demonstrate the effectiveness of the proposed scheme.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Teng Zhang, Tingchun Wang
Summary: Rigorous analysis was conducted on the error bound and conservation laws of a fourth-order compact finite difference scheme for the Zakharov system, providing error bounds for well-prepared and ill-prepared initial data, with high oscillation of solutions posing challenges in analyzing error bounds of numerical methods.
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Gengen Zhang, Chunmei Su
Summary: This paper proposes a uniformly accurate compact finite difference method to solve the quantum Zakharov system with a dimensionless parameter. The method considers the oscillatory initial layers and establishes error estimates that are uniform in both time and space. The numerical results verify the effectiveness of the method.
JOURNAL OF COMPUTATIONAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Yuyu He, Hongtao Chen
Summary: In this paper, an efficient and conservative compact difference scheme based on the scalar auxiliary variable (SAV) approach is proposed for solving the coupled Schrodinger-Boussinesq (CSB) equations. The scheme preserves the discrete modified energy and the convergent rates of second-order in time and fourth-order in space are proven using the discrete energy method. Numerical experiments are conducted to validate the theoretical analysis.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Yuyu He, Hongtao Chen
Summary: In this paper, an efficient and conservative compact difference scheme based on the SAV approach for the BP equation is constructed. The scheme preserves mass and discrete modified energy, with uniquely solvable properties and bounded estimates for the numerical solution. Convergence rates of second-order in temporal direction and fourth-order in spatial direction are detailed, with theoretical analysis verified through numerical experiments.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics
Cui Guo, Yinglin Wang, Yuesheng Luo
Summary: In this paper, a new numerical method using multiple integration to convert a partial differential equation into a pure integral equation is proposed for solving the nonlinear Rosenau-KdV equation. By avoiding the large errors caused by finite difference methods, the method shows high accuracy and conservativeness. The numerical results are consistent with analytical results, demonstrating the effectiveness of the discrete scheme.
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
Hongtao Chen, Yuyu He
Summary: In this paper, a conservative compact difference scheme for the generalized Kawahara equation is constructed based on the scalar auxiliary variable (SAV) approach. The discrete conservative laws of mass and Hamiltonian energy and boundedness estimates are studied in detail. The error estimates in discrete L-infinity norm and L-2 norm of the presented scheme are analyzed using mathematical induction and discrete energy method. An efficient algorithm, which only requires solving two decoupled equations, is proposed for the presented scheme.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Hong Cheng, Xiaofeng Wang
Summary: In this paper, a linear mass and energy conservative finite difference scheme for the generalized Korteweg-de Vries (GKdV) equation is proposed and analyzed. The scheme has second- and fourth-order accuracy in time and space, respectively, and is proven to be uniquely solvable, stable and convergent. Numerical examples confirm the stability and convergence of the scheme and its effectiveness in handling single and multi-solitary waves.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2023)
Article
Mathematics, Applied
Yuyu He, Xiaofeng Wang, Ruihua Zhong
Summary: In this paper, a novel three-point fourth-order compact operator is used to construct a new linearized conservative compact finite difference scheme for the symmetric regularized long wave equations. The discrete conservative laws, boundedness, and unique solvability are studied, and the convergence order and stability of the scheme are proved by the discrete energy method. Numerical examples are provided to support the theoretical analysis.
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2022)
Article
Mathematics, Interdisciplinary Applications
Ziyang Luo, Xingdong Zhang, Shuo Wang, Lin Yao
Summary: In this paper, a new numerical scheme is proposed to solve time fractional partial integro-differential equations with a weakly singular kernel. The proposed scheme, based on weighted and shifted Grunwald formula and compact difference operate, ensures stability and convergence, with accuracy independent of the fractional parameter a.
CHAOS SOLITONS & FRACTALS
(2022)
Article
Mathematics, Applied
Jianqiang Xie, Xiao Yan, Muhammad Aamir Ali, Zakia Hammouch
Summary: In this paper, an efficient physical-property-preserving algorithm is proposed and analyzed for the space fractional-order generalized Zakharov system. The system is reformulated as an equivalent system of equations by introducing the auxiliary equation. A spatial fourth-order physical-property-preserving linearly implicit difference scheme is developed for the transformed system. The scheme is proven to have the optimal order of O(Delta t(2) + h(4)) in discrete L-infinity and L-2 norms through the use of a cut-off function and discrete energy analysis method. The scheme is characterized by its physical-property preservation, linear decoupling, and suitability for parallel computing, especially in long time simulations and large-scale problems. Ample numerical results are provided to demonstrate the efficiency and preservation properties of the scheme, as well as investigate the dynamic behaviors of different solitary waves. (c) 2023 Elsevier B.V. All rights reserved.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Dingwen Deng, Qiang Wu
Summary: This paper investigates the numerical solutions of the nonlinear couple wave equations using the combination of compact difference method, Predictor-Corrector iterative methods, and Richardson extrapolation methods. The proposed methods are proven to be stable, convergent, and efficient through theoretical analysis and numerical experiments.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2023)
Article
Mathematics, Applied
Pradip Roul, V. M. K. Prasad Goura
Summary: This paper introduces a numerical technique for solving the time-fractional Black-Scholes equation governing European options, with a focus on stability, convergence, and the impact of fractional order derivative on option price profiles. The method's efficiency and accuracy are demonstrated through test problems with known analytical solutions, as well as its applicability to problems with unknown analytical solutions.
APPLIED NUMERICAL MATHEMATICS
(2021)
