Article
Mathematics, Applied
Yaping Li, Weidong Zhao, Wenju Zhao
Summary: This paper presents an efficient finite element method for the natural convection equations using the scalar auxiliary variable approach. The nonlinear terms in the Navier-Stokes equations and the heat equation are discretized using the linearly extrapolated Crank-Nicolson techniques. The induced scalar auxiliary equation is a univariate ordinary equation. The proposed method achieves redefined stability under the fully explicit scheme for the nonlinear terms without requiring skew-symmetric trilinear forms. The paper proves optimal convergence rates in space for all variables and also obtains second order convergence rates in time. Numerical experiments are conducted to support the theoretical analysis and demonstrate the efficiency of the proposed scheme.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Min Wang, Qiumei Huang, Cheng Wang
Summary: In this paper, a second order accurate numerical scheme for the square phase field crystal equation is proposed and analyzed, with the introduction of a 4-Laplacian term leading to higher nonlinearity. The scheme is made linear while preserving non-linear energy stability through the use of the scalar auxiliary variable (SAV) approach, with energy stability achieved by introducing an auxiliary variable and constant-coefficient diffusion terms with positive eigenvalues. The proposed method demonstrates efficiency and accuracy through numerical experiments.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Juan Zhang, Lixiu Dong, Zhengru Zhang
Summary: In this paper, a second-order accurate numerical scheme using the scalar auxiliary variable (SAV) method and Fourier-spectral method in space is proposed and analyzed for the droplet liquid film coarsening model. The scheme is linear and efficient, and it exhibits unconditional energy stability due to the application of SAV approach. A rigorous error estimate is provided, showing that the scheme with Fourier-spectral method converges with order O (t² + hm), where t and h are time and space step sizes, respectively. Numerical experiments confirm the efficiency and accuracy of the proposed scheme, including tests of convergence, mass conservation, and energy decrease. The simulation of coarsening process with time is also observed.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Dongfang Li, Xiaoxi Li, Hai-wei Sun
Summary: The paper reformulates the coupled nonlinear Schrodinger equation using the scalar auxiliary variable approach and solves it using Crank-Nicolson finite element method. The fully discrete method is shown to conserve mass and energy, but difficulties in analyzing numerical stability arise due to the presence of ut and vt terms in the equation. The challenges are technically overcome by analyzing the errors in the H-1 norm and connecting errors between the systems, leading to a convergent numerical solution at the order of O(tau 2+hp) in the H1 norm.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Jinliang Yan, Ling Zhu, Fuqiang Lu, Sihui Zheng
Summary: In this paper, linearly implicit modified energy-conserving schemes are proposed for the modified Korteweg-de Vries equation (mKdV) based on the invariant energy quadratization (IEQ) approach and the scalar auxiliary variable (SAV) approach. The Fourier pseudospectral method is used for spatial discretization, and Crank-Nicolson and Leap-Frog methods are used for temporal discretization. The conservation properties, existence and uniqueness, and linear stability of the proposed schemes are analyzed. The optimal order convergence rate of the semi-discrete scheme and the fully discrete schemes are also analyzed. Numerical examples are presented to illustrate the effectiveness of the proposed schemes.
NUMERICAL ALGORITHMS
(2022)
Article
Mathematics, Applied
Archna Kumari, Vijay Kumar Kukreja
Summary: In this study, a robust septic Hermite collocation method (SHCM) is proposed to simulate the Kuramoto-Sivashinsky (KS) equation. The algorithm is demonstrated to be unconditionally stable using the von-Neumann approach. Convergence analysis shows second-order convergent in the temporal direction and sixth-order convergent in the spatial direction. The proposed technique outperforms other methods and matches well with the analytical solution.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics
Taohua Liu, Xiucao Yin, Yinghao Chen, Muzhou Hou
Summary: In this paper, a practical numerical method for solving a one-dimensional two-sided space-fractional diffusion equation with variable coefficients in a finite domain is investigated. The method is based on the classical Crank-Nicolson (CN) method combined with Richardson extrapolation, and it provides second-order exact numerical estimates in time and space. The unconditional stability and convergence of the method are also tested, and two numerical examples are presented and compared with the exact solution.
Article
Mathematics, Applied
Yingwen Guo, Yinnian He
Summary: An efficient method is examined for the Oldroyd fluid, which reduces the nonlinear integro-differential equations to linear equations, significantly increasing computational efficiency. The approach uses finite element method in space and second-order Crank-Nicolson extrapolation in time. The method is unconditionally stable and convergent, with L-2 optimal error estimate and second-order accuracy. Numerical experiments demonstrate its accuracy, stability, and convergence.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Dianming Hou, Hui Wang, Chao Zhang
Summary: This paper proposes and analyzes positivity-preserving, unconditionally energy stable, and linear second order fully discrete schemes for Micro-Electromechanical system (MEMS). The numerical schemes use the first order backward difference formulation (BDF1) and second order Crank-Nicolson (CN) formulation for temporal discretization, and the central finite difference method for spatial discretization. The unconditional energy stability of the numerical schemes is rigorously proved. The numerical solutions always preserve the positivity property of the MEMS model and a series of numerical simulations demonstrate the positivity and energy stability of the proposed schemes.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Rui M. P. Almeida, Teofilo D. Chihaluca, Jose C. M. Duque
Summary: This paper investigates a class of nonlinear option pricing models considering transaction costs, focusing on numerical analysis of the Delta equation. A numerical algorithm for solving a generalized Black-Scholes partial differential equation in European option pricing is proposed, and the accuracy, convergence, and efficiency of different discretization methods are discussed. The proposed method's efficiency and accuracy are tested numerically, confirming theoretical behavior and good agreement with exact solutions.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Physics, Multidisciplinary
Chunya Wu, Xinlong Feng, Lingzhi Qian
Summary: In this paper, we propose a fully discrete and decoupled CNLF scheme for solving the MPFC model with long-range interaction. The CNLF scheme treats stiff terms implicitly with Crank-Nicolson and non-stiff terms explicitly with Leap-Frog, and combines with the SAV method to allow explicit treatment of the nonlinear potential. The scheme is shown to be fully discrete, second-order decoupled, and unconditionally stable, and is verified through numerical experiments in both 2D and 3D.
Article
Mathematics, Applied
Chunye Gong, Dongfang Li, Lili Li, Dan Zhao
Summary: Due to the nonlocality and nonlinearity, it is usually difficult to approximate and analyze the semi-linear parabolic differential equations with nonlocal initial conditions. Several researchers have presented different numerical schemes for solving the problems. Only second-order convergence results in spatial direction are proved so far. In order to effectively solve the problem, the problem is approximated by using the fourth order compact difference method in the spatial direction and the second-order Crank-Nicolson (CN) method in the temporal direction. The fully discrete numerical scheme gives a large system of nonlinear algebraic equations. The fixed-point iteration method with preconditioned GMRES is proposed for solving the resulting nonlinear systems. And the eigenvalues of the coefficient matrix are explored by the Hamilton-Cayley theorem and the investigation of the matrix structure. Stability and convergence of the fully discrete scheme are obtained. Numerical experiments are given to confirm the theoretical results.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Computer Science, Interdisciplinary Applications
Zengqiang Tan, Huazhong Tang
Summary: This paper studies linear and unconditionally modified-energy stable schemes for gradient flows. The schemes use general linear time discretizations and extrapolation to ensure stability. Numerical experiments are conducted to validate the stability of the schemes.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Maosheng Jiang, Zengyan Zhang, Jia Zhao
Summary: The scalar auxiliary variable (SAV) method is widely used in solving thermodynamically consistent PDE problems, but its numerical scheme raises the issue of preserving the energy law. This paper presents the relaxedSAV (RSAV) method, which overcomes this issue and improves accuracy and consistency.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Qianqian Chu, Guanghui Jin, Jihong Shen, Yuanfeng Jin
Summary: The numerical method implemented to solve the Allen-Cahn equation demonstrates unconditional convergence and preservation of the maximum principle without restrictions on step size, as verified through numerical experiments.
JOURNAL OF COMPUTATIONAL MATHEMATICS
(2021)
