Article
Mathematics, Applied
Hailiang Liu, Peimeng Yin
Summary: We propose an unconditionally energy stable Runge-Kutta discontinuous Galerkin scheme for solving a class of fourth order gradient flows, including the Swift-Hohenberg equation. Our algorithm achieves arbitrarily high order approximations in both space and time, while preserving energy dissipation for any time steps and spatial meshes. Numerical tests demonstrate the high order accuracy, energy stability, and simplicity of the proposed algorithm.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Zhengguang Liu, Chuanjun Chen
Summary: The Swift-Hohenberg model is a crucial phase field crystal model that can effectively describe diverse crystal phenomena. This study investigates several linear, second-order, and unconditionally energy stable schemes based on the traditional and modified scalar auxiliary variable (SAV) approaches. The newly proposed modified SAV approaches, namely step-by-step solving scalar auxiliary variable (3S-SAV) and exponential scalar auxiliary variable (E-SAV), are proven to possess several advantages over the traditional SAV method. The unconditional energy stability of all the semi-discrete schemes is carefully and rigorously demonstrated. Computational results indicate that the novel semi-implicit schemes save significant CPU time compared to the traditional SAV scheme. Finally, various 2D numerical simulations are presented to confirm the stability and accuracy of the proposed approaches.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Longzhao Qi, Yanren Hou
Summary: In this study, energy-stable linear schemes for solving the Swift-Hohenberg equation are proposed. The schemes are shown to satisfy the energy dissipation property through rigorous analysis and numerical validation, and error estimates are derived.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Hong Sun, Xuan Zhao, Haiyan Cao, Ran Yang, Ming Zhang
Summary: This paper discusses the design, analysis, and numerical simulations of a stabilized variable time-stepping difference scheme for the Swift-Hohenberg equation. The proposed scheme preserves a discrete energy dissipation law and achieves unique solvability and unconditional energy stability through new discrete orthogonal convolution kernels. Additionally, the proposed scheme demonstrates second-order L2 norm convergence in both time and space, independent of the time-step ratios. This is the first time L2 norm convergence of the adaptive BDF2 method is achieved for the Swift-Hohenberg equation.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2022)
Article
Mathematics, Applied
Longzhao Qi, Yanren Hou
Summary: This paper proposes and analyzes a second-order energy stable numerical scheme for the Swift-Hohenberg equation, utilizing mixed finite element approximation in space. The scheme employs a second-order backward differentiation formula scheme with a second-order stabilized term to guarantee long-term energy stability. It is proven to be unconditionally energy stable and uniquely solvable, with an optimal error estimate provided.Numerical experiments are conducted to support the theoretical analysis.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Yayun Fu, Dongdong Hu, Zhuangzhi Xu
Summary: The paper presents a class of high-order explicit exponential time differencing energy-preserving schemes for conservative fractional PDEs based on the general Hamiltonian form. By reformulating the equation into an equivalent system with a new quadratic energy and combining the projection technique, highly efficient schemes are obtained for stiff systems.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Kai Yang
Summary: This paper introduces a new class of numerical schemes for the generalized KdV equation, based on the scalar auxiliary variable method. By reformulating the equation using an auxiliary variable and utilizing Fourier pseudospectral spatial discretization, the scheme achieves exact conservation of momentum and energy, along with spectral accuracy in mass preservation.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Longzhao Qi, Yanren Hou
Summary: In this paper, we propose and analyze an unconditionally energy-stable, second-order-in-time, finite element scheme for the Swift-Hohenberg equation. We rigorously prove that our scheme is unconditionally solvable and energy stable. We also demonstrate the boundedness of discrete phase variable for any time and space mesh sizes. Numerical tests are conducted to validate the accuracy and energy stability of our scheme.
COMPUTATIONAL & APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Zhengguang Liu
Summary: The Swift-Hohenberg model is an important phase field crystal model with quadratic-cubic nonlinearity, presenting challenges due to negative energy for energy stability. This paper introduces two energy stable schemes and proves the unconditional energy stability of all semi-discrete schemes. Various 2D numerical simulations are used to demonstrate stability and accuracy.
NUMERICAL ALGORITHMS
(2021)
Article
Mathematics, Applied
Kai Yang
Summary: This paper proposes a new class of conservative numerical schemes for the generalized KdV equation, based on the scalar auxiliary variable method. It reformulates the equation and preserves the energy and momentum invariants using the quadratic preserving Runge-Kutta method. Numerical experiments demonstrate the efficiency of this scheme in simulating breathers for the modified KdV equation.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2022)
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
Mathematics, Applied
Lingling Zhou, Ruihan Guo
Summary: This paper presents a local discontinuous Galerkin (LDG) method for the Swift-Hohenberg equation and provides numerical tests and theoretical analysis, demonstrating its stability and superior error estimation.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Seunggyu Lee, Sungha Yoon, Junseok Kim
Summary: This paper examines the effective temporal step size for convex splitting schemes in simulating the Swift-Hohenberg equation. It presents specific formulations for different methods and verifies their effectiveness through numerical simulations.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Materials Science, Multidisciplinary
Israr Ahmad, Thabet Abdeljawad, Ibrahim Mahariq, Kamal Shah, Nabil Mlaiki, Ghaus Ur Rahman
Summary: This article presents an approximate analytical solution to the fractional order Swift-Hohenberg equation using a novel iterative method called Laplace Adomian decomposition method (LADM). Through this method, nonlinear FSH equations with and without dispersive terms are studied and results are compared through plots for different fractional orders. The article also explores the problem under Caputo-Fabrizio fractional order derivative (CFFOD) and provides examples to demonstrate the results.
RESULTS IN PHYSICS
(2021)
Article
Mathematics, Applied
Jingying Wang, Chen Cui, Zhifeng Weng, Shuying Zhai
Summary: This paper presents a high order time discretization method by combining the time splitting scheme with semi-implicit spectral deferred correction method for the space-fractional Swift-Hohenberg equation (SFSH). The stability and convergence of the obtained numerical scheme are analyzed theoretically. Various numerical experiments are performed to validate the theoretical results and efficiency of the proposed method.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
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)