Article
Mechanics
T. Kossaczka, M. Ehrhardt, M. Guenther
Summary: In this paper, a new modification of the weighted essentially non-oscillatory (WENO) method for solving nonlinear degenerate parabolic equations is developed using deep learning techniques. The modified WENO-DS method, trained with a convolutional neural network, maintains consistency and convergence while outperforming the standard WENO method in handling sharp interfaces and providing good resolution of discontinuities.
Article
Computer Science, Interdisciplinary Applications
Deniz A. Bezgin, Steffen J. Schmidt, Nikolaus A. Adams
Summary: Neural networks have been integrated into the WENO scheme to address challenges in achieving maximum-order convergence and ENO property, demonstrating good generalizability and performance in various test cases. The WENO3-NN scheme learns a non-trivial dispersion-dissipation relation and may introduce vanishing dissipation near the cutoff wavenumber, which is counterintuitive to classical discretization-design principles.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Ki Wai Fong, Shingyu Leung
Summary: This article introduces two new interpolation methods for S-2. In the first part, a simple interpolation method called Spherical Interpolation of orDER n (SIDER-n) is presented, which provides a Cn interpolant for n ≥ 2. This method generalizes the construction of Bezier curves developed for R. The second part incorporates the ENO philosophy and develops a new Spherical Essentially Non-Oscillatory (SENO) interpolation method. This approach can reduce spurious oscillations in high-order reconstruction when the underlying curve on S-2 has kinks or sharp discontinuity in the higher derivatives. Multiple examples are provided to demonstrate the accuracy and effectiveness of the proposed approaches.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics
Omer Musa, Guoping Huang, Mingsheng Wang
Summary: The paper proposes a new simple smoothness indicator for fifth-order linear reconstruction, reducing the complexity of the WENO-AO(5,3) scheme. It modifies the WENO-AO(5,3) scheme to the WENO-O scheme with a new and simple formulation, showing through numerical experiments its accuracy and efficacy compared to original schemes. The results indicate that the proposed WENO-O scheme is not only comparable in accuracy and efficacy to the original scheme but also decreases computational cost and complexity.
Article
Mathematics, Applied
Shujiang Tang, Yujie Feng, Mingjun Li
Summary: In this paper, we have improved the classical WEND JS and WENO-Z schemes by constructing a selector that can identify the less-smooth sub-stencils. We have also developed two new WENO schemes, WENO-IJS and WENO-IZ, which can adaptively increase the weight of less-smooth sub-stencils. Theoretical analysis and numerical experiments show that these schemes maintain the non-oscillatory (ENO) property, have lower numerical dissipation at discontinuities, and exhibit better spectral characteristics compared to other schemes.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Computer Science, Interdisciplinary Applications
Rongqian Chen, Linkuan Wu, Yancheng You
Summary: In this paper, a new type of high order weighted compact scheme based on ZJS and ZQ methods is designed, which shows better resolution, smaller dissipation, and the ability to capture strong shock waves through comparisons with other schemes.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mechanics
Jianguo Ning, Xuan Su, Xiangzhao Xu
Summary: This study proposes a modified WENO scheme for numerical simulation of complex compressible flow fields. By introducing local and global smoothness indicators and constructing an adaptive coefficient, this scheme improves accuracy, reduces dissipation, and prevents spurious numerical oscillations.
Article
Computer Science, Interdisciplinary Applications
Andrea Di Mascio, Stefano Zaghi
Summary: A new immersed boundary approach for high order WENO schemes is proposed, which combines ideas from general immersed boundary algorithms and the level-set approach. Despite being formally second order accurate, numerical tests show that using higher order approximation for Eulerian fluxes can conveniently capture flow details and reduce uncertainty even with very coarse grids.
COMPUTERS & FLUIDS
(2021)
Article
Mathematics, Applied
Zhenming Wang, Linlin Tian, Jun Zhu, Ning Zhao
Summary: In this paper, a hybrid unequal-sized weighted essentially non-oscillatory (US-WENO) scheme is developed to reduce the computational cost. The proposed hybridization strategy can automatically and efficiently identify the troubled cells, and does not contain artificial parameters. Numerical experiments show that the proposed hybrid method can inherit all the features of the existing US-WENO scheme while improving its computational efficiency.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Mathematics, Applied
H. Carrillo, C. Pares, D. Zorio
Summary: The goal of this work is to introduce new families of shock-capturing high-order numerical methods for systems of conservation laws that combine Fast WENO and Optimal WENO reconstructions with Approximate Taylor methods for the time discretization. These new methods are compared between them and against methods based on standard WENO implementations and/or SSP-RK time discretization. Various test cases are considered to evaluate the performance of the new methods.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Indra Wibisono, Yanuar, Engkos A. Kosasih
Summary: The TENO scheme presented in this study utilizes Hermite polynomials for efficient and targeted non-oscillatory reconstruction, incorporating compact reconstruction and low dissipation advantages. It introduces a new high-order global smoothness indicator and demonstrates improved shock-capturing performance in numerical tests.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Yahui Wang
Summary: In this article, a new modified stencil approximation method for reducing numerical dissipation of classical weighted essentially non-oscillatory (WENO-JS) schemes is proposed. The method improves the accuracy of approximation polynomials and calculates candidate fluxes, achieving optimal convergence order in smooth regions. Numerical examples demonstrate that the proposed WENO-MS schemes provide comparable or higher resolution compared with existing schemes.
COMPUTATIONAL & APPLIED MATHEMATICS
(2023)
Article
Mechanics
Zhenming Wang, Jun Zhu, Chunwu Wang, Ning Zhao
Summary: This paper proposes two unequal-sized weighted essentially non-oscillatory (US-WENO) schemes for solving hyperbolic conservation laws. The first scheme, called alternative US-WENO (AUS-WENO), is based on the values of conserved variables at the grid points and has smaller numerical errors. The second scheme, hybrid AUS-WENO, improves computational efficiency by combining a hybrid strategy.
Article
Computer Science, Interdisciplinary Applications
Shiyao Li, Yiqing Shen, Ke Zhang, Ming Yu
Summary: In this paper, a novel weighting method is proposed to improve the accuracy of the higher-order WENO-ZN scheme at critical points. The method is extended to construct higher-order WENO-ZN schemes, and numerical experiments demonstrate their effectiveness in capturing shock waves and improving accuracy in smooth regions.
COMPUTERS & FLUIDS
(2022)
Article
Computer Science, Interdisciplinary Applications
Mojtaba Balaj, Mohammad Hassan Djavareshkian
Summary: A pressure-based semi-implicit procedure has been developed for the computation of compressible flows, showing reliable results in shock tube and different Mach numbers. The numerical method is also used to investigate flow under various conditions, resulting in significant improvements.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS
(2021)
Article
Mathematics, Applied
Mostafa Abbaszadeh, Mahmoud A. Zaky, Ahmed S. Hendy, Mehdi Dehghan
Summary: In this paper, a numerical formulation with second-order accuracy in the time direction and spectral accuracy in the space variable is proposed for solving a nonlinear high-dimensional Rosenau-Burgers equation. The spectral element method and the two-grid idea are combined to simulate the equation, and a three-level algorithm is used for the proposed technique. The existence and uniqueness of the solutions to Steps 1, 2, and 3 are investigated, and error analysis is also discussed.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Mostafa Abbaszadeh, Mohammad Ivan Azis, Mehdi Dehghan, Reza Mohammadi-Arani
Summary: This paper proposes a new meshless numerical procedure, namely the gradient smoothing method (GSM), for simulating the pollutant transition equation in urban street canyons. The time derivative is approximated using the finite difference scheme, while the space derivative is discretized using the gradient smoothing method. Additionally, the proper orthogonal decomposition (POD) approach is employed to reduce CPU time. Several real-world examples are solved to verify the efficiency of the developed numerical procedure.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Biology
Niusha Narimani, Mehdi Dehghan
Summary: This paper numerically studies the therapies of prostate cancer in a two-dimensional space. The proposed model describes the tumor growth driven by a nutrient and the effects of cytotoxic chemotherapy and antiangiogenic therapy. The results obtained without using any adaptive algorithm show the response of the prostate tumor growth to different therapies.
COMPUTERS IN BIOLOGY AND MEDICINE
(2023)
Article
Engineering, Multidisciplinary
Mostafa Abbaszadeh, Yasmin Kalhor, Mehdi Dehghan, Marco Donatelli
Summary: The purpose of this research is to develop a numerical method for option pricing in jump-diffusion models. The proposed model consists of a backward partial integro-differential equation with diffusion and advection factors. Pseudo-spectral technique and cubic B-spline functions are used to solve the equation, and a second-order Strong Stability Preserved Runge-Kutta procedure is adopted. The efficiency and accuracy of the proposed method are demonstrated through various test cases.
ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS
(2023)
Article
Engineering, Multidisciplinary
Mahboubeh Najafi, Mehdi Dehghan
Summary: In this work, two-dimensional dendritic solidification is simulated using the meshless Diffuse Approximate Method (DAM). The Stefan problem is studied through the phase-field model, considering both isotropic and anisotropic materials for comparisons. The effects of changing some constants on the obtained patterns are investigated.
ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS
(2023)
Article
Mathematics, Applied
Rooholah Abedian, Mehdi Dehghan
Summary: This paper presents a new formulation of conservative finite difference radial basis function weighted essentially non-oscillatory (WENO-RBF) schemes to solve conservation laws. Unlike previous methods, the flux function is generated directly with the conservative variables, and arbitrary monotone fluxes can be employed. Numerical simulations of several benchmark problems are conducted to demonstrate the good performance of the new scheme.
ADVANCES IN APPLIED MATHEMATICS AND MECHANICS
(2023)
Article
Mathematics, Applied
Majid Haghi, Mohammad Ilati, Mehdi Dehghan
Summary: In this paper, the cubic-quintic complex Ginzburg-Landau (CQCGL) equation is numerically studied in 1D, 2D, and 3D spaces. The equation is decomposed into three subproblems using the Strang splitting technique. Nonlinear ODEs are solved by the Runge-Kutta technique for the first and third problems, while a fourth-order RBF-generated Hermite finite difference (RBF-HFD) method is used for the second problem involving spatial derivatives. A temporal Richardson extrapolation technique is applied to improve the order of convergence in the time direction. Numerical results show that the proposed method improves the order of convergence and is accurate and efficient.
COMPUTATIONAL & APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Alireza Hosseinian, Pouria Assari, Mehdi Dehghan
Summary: This paper presents a numerical method for solving nonlinear Volterra integral equations with delay arguments. The method uses the discrete collocation approach with thin plate splines as a type of radial basis functions. The method provides an effective and stable algorithm to estimate the solution, which can be easily implemented on a personal computer. The error analysis and convergence validation of the method are also provided.
COMPUTATIONAL & APPLIED MATHEMATICS
(2023)
Article
Engineering, Multidisciplinary
Fatemeh Asadi-Mehregan, Pouria Assari, Mehdi Dehghan
Summary: This paper presents a computational algorithm for solving nonlinear systems of ordinary and partial differential equations resulting from HIV infection models. The method uses local radial basis functions as shape functions in the discrete collocation scheme, approximating the solution by a small set of nodes. The computational efficiency of the scheme is studied through several test examples.
ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS
(2023)
Article
Engineering, Multidisciplinary
Ali Ebrahimijahan, Mehdi Dehghan, Mostafa Abbaszadeh
Summary: In this study, the integrated radial basis functions-partition of unity (IRBF-PU) method is proposed for solving the coupled Schrodinger-Boussinesq equations in one-and two-dimensions. The IRBF-PU method is a local mesh-free method that offers flexibility and high accuracy for PDEs with smooth initial conditions. Numerical simulations demonstrate that the IRBF-PU method can effectively simulate solitary waves and preserve conservation laws. Furthermore, the obtained results are compared with other methods in the literature to validate the effectiveness and reliability of the proposed method.
ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS
(2023)
Article
Engineering, Multidisciplinary
Hasan Zamani-Gharaghoshi, Mehdi Dehghan, Mostafa Abbaszadeh
Summary: This article presents a numerical method for solving the surface Allen-Cahn model. The method is based on the generalized moving least-squares approximation and the closest point method. It does not depend on the structure of the underlying surface and only requires a set of arbitrarily distributed mesh-free points on the surface.
ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS
(2023)
Article
Computer Science, Interdisciplinary Applications
Mostafa Abbaszadeh, Mehdi Dehghan, Dunhui Xiao
Summary: This paper presents a new numerical formulation for simulating tumor growth. The proposed method utilizes the meshless Galerkin technique and a two-grid algorithm to improve accuracy and efficiency in obtaining simulation results.
ENGINEERING WITH COMPUTERS
(2023)
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
Mostafa Abbaszadeh, Alireza Bagheri Salec, Taghreed Abdul-Kareem Hatim Aal-Ezirej
Summary: In this paper, an improved Boussinesq model is studied. The existence, uniqueness, stability and convergence of the solution are analyzed through discretization and finite difference methods. The proposed scheme is validated through examples in 1D and 2D cases.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Computer Science, Interdisciplinary Applications
Mostafa Abbaszadeh, AliReza Bagheri Salec, Shurooq Kamel Abd Al-Khafaji
Summary: This paper proposes a numerical method using spectral collocation and POD approach to solve systems of space fractional PDEs. The method achieves high accuracy and computational efficiency.
ENGINEERING COMPUTATIONS
(2023)
Article
Mathematics, Applied
Hao Liu, Yuzhe Li
Summary: This paper investigates the finite-time stealthy covert attack on reference tracking systems with unknown-but-bounded noises. It proposes a novel finite-time covert attack method that can steer the system state into a target set within a finite time interval while being undetectable.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Nikolay A. Kudryashov, Aleksandr A. Kutukov, Sofia F. Lavrova
Summary: The Chavy-Waddy-Kolokolnikov model with dispersion is analyzed, and new properties of the model are studied. It is shown that dispersion can be used as a control mechanism for bacterial colonies.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Qiang Ma, Jianxin Lv, Lin Bi
Summary: This paper introduces a linear stability equation based on the Boltzmann equation and establishes the relationship between small perturbations and macroscopic variables. The numerical solutions of the linear stability equations based on the Boltzmann equation and the Navier-Stokes equations are the same under the continuum assumption, providing a theoretical foundation for stability research.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Samuel W. Akingbade, Marian Gidea, Matteo Manzi, Vahid Nateghi
Summary: This paper presents a heuristic argument for the capacity of Topological Data Analysis (TDA) to detect critical transitions in financial time series. The argument is based on the Log-Periodic Power Law Singularity (LPPLS) model, which characterizes financial bubbles as super-exponential growth (or decay) with increasing oscillations approaching a tipping point. The study shows that whenever the LPPLS model fits the data, TDA generates early warning signals. As an application, the approach is illustrated using positive and negative bubbles in the Bitcoin historical price.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Xavier Antoine, Jeremie Gaidamour, Emmanuel Lorin
Summary: This paper is interested in computing the ground state of nonlinear Schrodinger/Gross-Pitaevskii equations using gradient flow type methods. The authors derived and analyzed Fractional Normalized Gradient Flow methods, which involve fractional derivatives and generalize the well-known Normalized Gradient Flow method proposed by Bao and Du in 2004. Several experiments are proposed to illustrate the convergence properties of the developed algorithms.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Lianwen Wang, Xingyu Wang, Zhijun Liu, Yating Wang
Summary: This contribution presents a delayed diffusive SEIVS epidemic model that can predict and quantify the transmission dynamics of slowly progressive diseases. The model is applied to fit pulmonary tuberculosis case data in China and provides predictions of its spread trend and effectiveness of interventions.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Shuangxi Huang, Feng-Fei Jin
Summary: This paper investigates the error feedback regulator problem for a 1-D wave equation with velocity recirculation. By introducing an invertible transformation and an adaptive error-based observer, an observer-based error feedback controller is constructed to regulate the tracking error to zero asymptotically and ensure bounded internal signals.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Weimin Liu, Shiqi Gao, Feng Xu, Yandong Zhao, Yuanqing Xia, Jinkun Liu
Summary: This paper studies the modeling and consensus control of flexible wings with bending and torsion deformation, considering the vibration suppression as well. Unlike most existing multi-agent control theories, the agent system in this study is a distributed parameter system. By considering the mutual coupling between the wing's deformation and rotation angle, the dynamics model of each agent is expressed using sets of partial differential equations (PDEs) and ordinary differential equations (ODEs). Boundary control algorithms are designed to achieve control objectives, and it is proven that the closed-loop system is asymptotically stable. Numerical simulation is conducted to demonstrate the effectiveness of the proposed control scheme.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Gourav Mandal, Lakshmi Narayan Guin, Santabrata Chakravarty
Summary: The ecological framework investigates the dynamical complexity of a system influenced by prey refuge and alternative food sources for predators. This study provides a thorough investigation of the stability-instability phenomena, system parameters sensitivity, and the occurrence of bifurcations. The bubbling phenomenon, which indicates a change in the amplitudes of successive cycles, is observed in the current two-dimensional continuous system. The controlling system parameter for the bubbling phenomena is found to be the most sensitive. The prediction and identification of bifurcations in the dynamical system are crucial for theoretical and field researchers.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Damian Trofimowicz, Tomasz P. Stefanski, Jacek Gulgowski, Tomasz Talaska
Summary: This paper presents the application of control engineering methods in modeling and simulating signal propagation in time-fractional electrodynamics. By simulating signal propagation in electromagnetic media using Maxwell's equations with fractional-order constitutive relations in the time domain, the equations in time-fractional electrodynamics can be considered as a continuous-time system of state-space equations in control engineering. Analytical solutions are derived for electromagnetic-wave propagation in the time-fractional media based on state-transition matrices, and discrete time zero-order-hold equivalent models are developed and their analytical solutions are derived. The proposed models yield the same results as other reference methods, but are more flexible in terms of the number of simulation scenarios that can be tackled due to the application of the finite-difference scheme.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Yuhao Zhao, Fanhao Guo, Deshui Xu
Summary: This study develops a vibration analysis model of a nonlinear coupling-layered soft-core beam system and finds that nonlinear coupling layers are responsible for the nonlinear phenomena in the system. By using reasonable parameters for the nonlinear coupling layers, vibrations in the resonance regions can be reduced and effective control of the vibration energy of the soft-core beam system can be achieved.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
S. Kumar, H. Roy, A. Mitra, K. Ganguly
Summary: This study investigates the nonlinear dynamic behavior of bidirectional functionally graded plates (BFG) and unidirectional functionally graded plates (UFG). Two different methods, namely the whole domain method and the finite element method, are used to formulate the dynamic problem. The results show that all three plates exhibit hardening type nonlinearity, with the effect of material gradation parameters being more pronounced in simply supported plates.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Isaac A. Garcia, Susanna Maza
Summary: This paper analyzes the role of non-autonomous inverse Jacobi multipliers in the problem of nonexistence, existence, localization, and hyperbolic nature of periodic orbits of planar vector fields. It extends and generalizes previous results that focused only on the autonomous or periodic case, providing novel applications of inverse Jacobi multipliers.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Yongjian Liu, Yasi Lu, Calogero Vetro
Summary: This paper introduces a new double phase elliptic inclusion problem (DPEI) involving a nonlinear and nonhomogeneous partial differential operator. It establishes the existence and extremality results to the elliptic inclusion problem and provides definitions for weak solutions, subsolutions, and supersolutions.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Shangshuai Li, Da-jun Zhang
Summary: In this paper, the Cauchy matrix structure of the spin-1 Gross-Pitaevskii equations is investigated. A 2 x 2 matrix nonlinear Schrodinger equation is derived using the Cauchy matrix approach, serving as an unreduced model for the spin-1 BEC system with explicit solutions. Suitable constraints are provided to obtain reductions for the classical and nonlocal spin-1 GP equations and their solutions, including one-soliton solution, two-soliton solution, and double-pole solution.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
