Article
Mathematics, Applied
Fu-Rong Lin, Yi-Feng Qiu, Zi-Hang She
Summary: This paper develops high order numerical schemes for solving the initial boundary value problem of one-dimensional and two-dimensional space fractional diffusion equations. The proposed IRK-WSGD schemes exhibit stable behavior with fourth order accuracy in time and second/third order accuracy in space. Preconditioning methods for linear systems are discussed, and numerical experiments confirm the accuracy and efficiency of the method.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Mathematics, Applied
Yuan Xu, Siu-Long Lei, Hai-Wei Sun
Summary: The anisotropic space-fractional diffusion equations in two dimensions are discretized using the Crank-Nicolson difference scheme with the weighted and shifted Grunwald formula. The coefficient matrix of the discretized linear system possesses a two-level Toeplitz-like structure. By utilizing the GMRES method with a newly proposed tridiagonal preconditioner as a smoother, the convergence rate of the multigrid method can be significantly accelerated.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Interdisciplinary Applications
Changqing Yang
Summary: In this study, we improved the spectral deferred correction method for solving nonlinear fractional differential equations. By transforming the problems into equivalent nonlinear Volterra integral equations with weakly singular kernels, we applied the fractional Adams-Bashforth method and the Gauss quadrature formula with fractional Adams-Moulton scheme to predict and correct the solutions. We also conducted a rigorous error analysis for the numerical scheme and provided computational results to demonstrate its accuracy and ease of implementation.
CHAOS SOLITONS & FRACTALS
(2023)
Article
Mathematics, Applied
Bo Tang, Yanping Chen, Xiuxiu Lin
Summary: In this paper, a space-time spectral Galerkin method is re-examined for solving multi-term time fractional diffusion equations, with improved a posteriori error estimates proposed and validated through numerical examples to confirm the effectiveness of the theoretical claims.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Mathematics, Applied
Wei Qu, Zhi Li
Summary: This study proposes a numerical solution based on the CN-ADI-FV discretization method, using the PCG method as an acceleration technique, which has been verified to achieve superlinear convergence in O(N log N) operations and is only used twice in the entire solution process.
APPLIED MATHEMATICS AND COMPUTATION
(2021)
Article
Mathematics, Applied
Fazal Ghaffar, Saif Ullah, Noor Badshah, Najeeb Alam Khan
Summary: In this paper, a higher-order compact finite difference scheme with multigrid algorithm is used to solve the one-dimensional time fractional diffusion equation. The scheme achieves eighth-order accuracy in space, and its convergence is proven through Fourier analysis and matrix analysis. Numerical experiments confirm the performance and accuracy of the proposed scheme.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics
Xin-Hui Shao, Chong-Bo Kang
Summary: This paper focuses on the discretized linear systems of the spatial fractional diffusion equations. We propose a modified DTS iteration method and provide its asymptotic convergence conditions. Furthermore, we design a fast modified DTS preconditioner by replacing Toeplitz matrix T with the tau matrix to accelerate the convergence rates of the GMRES method. Theoretical analysis shows that the spectrum of the fast modified DTS preconditioned matrix clusters around one. Numerical experiments validate the effectiveness of the constructed fast modified DTS preconditioner for GMRES method.
Article
Mathematics, Applied
Lisandro A. Raviola, Mariano F. De Leo
Summary: We evaluated the performance of novel numerical methods for solving one-dimensional nonlinear fractional dispersive and dissipative evolution equations and showed that the proposed methods are effective in terms of accuracy and computational cost. They can be applied to both irreversible models and dissipative solitons, offering a promising alternative for solving a wide range of evolutionary partial differential equations.
APPLIED MATHEMATICS AND COMPUTATION
(2024)
Article
Computer Science, Interdisciplinary Applications
Ramy M. Hafez, Magda Hammad, Eid H. Doha
Summary: This paper proposes two efficient Galerkin spectral algorithms for solving multi-dimensional time fractional advection-diffusion-reaction equations. The algorithms do not require imposition of artificial smoothness assumptions in time direction and the numerical results demonstrate their flexibility and effectiveness.
ENGINEERING WITH COMPUTERS
(2022)
Article
Physics, Fluids & Plasmas
Mario I. Molina
Summary: The study focused on linear and nonlinear modes of a one-dimensional nonlinear electrical lattice with a fractional discrete Laplacian. Long-range intersite coupling was induced by the fractional discrete Laplacian. In the linear regime, plane waves spectrum and mean-square displacement were computed in closed form, showing ballistic behavior at long times. In the nonlinear regime, the number of generated discrete solitons decreased as the fractional exponent decreased.
Article
Optics
Cristian Mejia-Cortes, Mario Molina
Summary: In this study, we investigate the existence and stability of nonlinear discrete vortex solitons in a square lattice by replacing the standard discrete Laplacian with a fractional version. A new effective site-energy term and a coupling among sites are created, with the range of coupling becoming effectively long range at small alpha values. It is observed that the stability domain of discrete vortex solitons is extended to lower power levels as the alpha coefficient diminishes.
Article
Mathematics, Applied
Muhammad Hamid, Muhammad Usman, Wei Wang, Zhenfu Tian
Summary: The study introduces an operational matrix-based spectral computational method coupled with the Picard technique for seeking solutions to a family of nonlinear evolution differential equations, proving to be more accurate and efficient than existing methods.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
Maohua Ran, Xiaojuan Lei
Summary: This paper focuses on the fast computation for solving the fourth-order time fractional diffusion wave equations with variable coefficient, deriving a fast difference scheme that significantly reduces computational complexity while maintaining almost the same accuracy as traditional schemes. The solvability, unconditional stability, and convergence of the scheme are rigorously proved using the discrete energy method, with numerical results provided to verify its performance.
APPLIED NUMERICAL MATHEMATICS
(2021)
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)
Article
Optics
Mario I. Molina
Summary: In this study, we analytically and numerically examine the influence of fractionality on a saturable bulk and surface impurity embedded in a one-dimensional lattice. We use a previously introduced fractional Laplacian and lattice Green functions to obtain bound state energies and amplitude profiles as functions of the fractional exponent s and saturable impurity strength x. The transmission is expressed as a function of s and x, showing significant deviations from the standard case at small fractional exponent values. The self-trapping of initially localized excitation is qualitatively similar for both the bulk and surface mode, but complete confinement is achieved at s -> 0, as demonstrated theoretically and observed numerically.
Article
Mathematics
Manuel Bogoya, Sergei Grudsky, Mariarosa Mazza, Stefano Serra-Capizzano
Summary: This article provides a detailed analysis of the spectral features of a Toeplitz matrix-sequence generated by a real-valued function f and investigates the conditioning of the matrices when f is nonnegative. In the context of numerical approximation of distributed-order fractional differential equations (FDEs), a novel type of problem is considered where the matrices have a specific form. Selected numerical experiments are presented to verify the theoretical analysis and discuss open questions and future investigations.
LINEAR & MULTILINEAR ALGEBRA
(2023)
Article
Mathematics, Applied
Mariarosa Mazza, Marco Donatelli, Carla Manni, Hendrik Speleers
Summary: In this work, we study a fractional differential equation in Riesz form discretized by a polynomial B-spline collocation method. The resulting coefficient matrices possess a Toeplitz-like structure and their spectral properties are investigated. We prove that these matrices are ill-conditioned in both the low and high frequencies for large polynomial degrees. Moreover, we find similarities between our problem and classical diffusion problems.
NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS
(2023)
Article
Computer Science, Interdisciplinary Applications
Armando Coco, Mariarosa Mazza, Matteo Semplice
Summary: We propose a Boundary Local Fourier Analysis (BLFA) method to optimize the relaxation parameters of boundary conditions in a multigrid framework. The method is designed for solving elliptic equations on curved domains and can be extended to general PDEs in curved domains. The boundary is implicitly defined by a level-set function and a ghost-point technique is used to handle the boundary conditions. The relaxation parameters are optimized based on the distance between ghost points and boundary to smooth the residual along the tangential direction.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics, Applied
Rafael Diaz Fuentes, Marco Donatelli, Caterina Fenu, Giorgio Mantica
Summary: This paper reviews several methods for approximating the trace of a symmetric matrix Omega and proposes a block stochastic method. The results of this technique converge quickly and it has the same computational advantages as the partial global Lanczos method.
NUMERICAL ALGORITHMS
(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
Computer Science, Interdisciplinary Applications
Hasan Zamani-Gharaghoshi, Mehdi Dehghan, Mostafa Abbaszadeh
Summary: This paper presents a local meshless collocation method for solving reaction-diffusion systems on surfaces. The proposed numerical procedure utilizes Pascal polynomial approximation and closest point method. This method is geometrically flexible and can be used to solve partial differential equations on unstructured point clouds. It only requires a set of arbitrarily scattered mesh-free points representing the underlying surface.
ENGINEERING WITH COMPUTERS
(2023)
Article
Computer Science, Interdisciplinary Applications
Reza MohammadiArani, Mehdi Dehghan, Mostafa Abbaszadeh
Summary: Different coupled systems for the shallow water equation, bed elevation, and suspended load equation have been proposed. The main goal of this paper is to utilize an advanced lattice Boltzmann method (LBM) to solve this system of equations. In addition, a practical approach is developed for applying open boundary conditions in order to relax the solution onto a prescribed equilibrium flow.
ENGINEERING WITH COMPUTERS
(2023)
Article
Mathematics, Applied
Mehdi Dehghan, Zeinab Gharibi
Summary: This paper discusses the incompressible miscible displacement of two-dimensional Darcy-Forchheimer flow and formulates a mathematical model with two partial differential equations: a Darcy-Forchheimer flow equation for the pressure and a convection-diffusion equation for the concentration. The model is discretized using a fully mixed virtual element method (VEM) and stability, existence, and uniqueness of the associated mixed VEM solution are proved under smallness data assumption. Optimal error estimates are obtained for concentration, auxiliary flux variables, and velocity, and several numerical experiments are presented to support the theoretical analysis and illustrate the applicability for solving actual problems.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2023)
Article
Mathematics, Applied
Fatemeh Asadi-Mehregan, Pouria Assari, Mehdi Dehghan
Summary: In this research, we present a numerical approach for solving a specific type of nonlinear integro-differential equations derived from Volterra's population model. This model captures the growth of a biological species in a closed system and includes an integral term to account for toxin accumulation. The proposed technique utilizes the discrete Galerkin scheme with the moving least squares (MLS) algorithm to estimate the solution of the integro-differential equations.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(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
Mathematics
D. Ahmad, M. Donatelli, M. Mazza, S. Serra-Capizzano, K. Trotti
Summary: This article discusses the numerical solution of time-dependent space tempered fractional diffusion equations. The use of Crank-Nicolson in time and second-order accurate tempered weighted and shifted Grunwald difference in space leads to dense Toeplitz-like linear systems. By exploiting the related structure, a specialized multigrid solver and multigrid-based preconditioners are designed, all with weighted Jacobi as a smoother. A new smoothing analysis is provided, which expands the set of suitable Jacobi weights and confirms the computational effectiveness of the resulting multigrid-based solvers for tempered fractional diffusion equations.
LINEAR & MULTILINEAR ALGEBRA
(2023)
Article
Computer Science, Interdisciplinary Applications
Tian Liang, Lin Fu
Summary: In this work, a new shock-capturing framework is proposed based on a new candidate stencil arrangement and the combination of infinitely differentiable non-polynomial RBF-based reconstruction in smooth regions with jump-like non-polynomial interpolation for genuine discontinuities. The resulting scheme achieves high order accuracy and resolves genuine discontinuities with sub-cell resolution.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Lukas Lundgren, Murtazo Nazarov
Summary: In this paper, a high-order accurate finite element method for incompressible variable density flow is introduced. The method addresses the issues of saddle point system and stability problem through Schur complement preconditioning and artificial compressibility approaches, and it is validated to have high-order accuracy for smooth problems and accurately resolve discontinuities.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Gabriele Ciaramella, Laurence Halpern, Luca Mechelli
Summary: This paper presents a novel convergence analysis of the optimized Schwarz waveform relaxation method for solving optimal control problems governed by periodic parabolic PDEs. The analysis is based on a Fourier-type technique applied to a semidiscrete-in-time form of the optimality condition, which enables a precise characterization of the convergence factor at the semidiscrete level. The behavior of the optimal transmission condition parameter is also analyzed in detail as the time discretization approaches zero.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Jonas A. Actor, Xiaozhe Hu, Andy Huang, Scott A. Roberts, Nathaniel Trask
Summary: This article introduces a scientific machine learning framework that uses a partition of unity architecture to model physics through control volume analysis. The framework can extract reduced models from full field data while preserving the physics. It is applicable to manifolds in arbitrary dimension and has been demonstrated effective in specific problems.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Nozomi Magome, Naoki Morita, Shigeki Kaneko, Naoto Mitsume
Summary: This paper proposes a novel strategy called B-spline based SFEM to fundamentally solve the problems of the conventional SFEM. It uses different basis functions and cubic B-spline basis functions with C-2-continuity to improve the accuracy of numerical integration and avoid matrix singularity. Numerical results show that the proposed method is superior to conventional methods in terms of accuracy and convergence.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Timothy R. Law, Philip T. Barton
Summary: This paper presents a practical cell-centred volume-of-fluid method for simulating compressible solid-fluid problems within a pure Eulerian setting. The method incorporates a mixed-cell update to maintain sharp interfaces, and can be easily extended to include other coupled physics. Various challenging test problems are used to validate the method, and its robustness and application in a multi-physics context are demonstrated.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Xing Ji, Fengxiang Zhao, Wei Shyy, Kun Xu
Summary: This paper presents the development of a third-order compact gas-kinetic scheme for compressible Euler and Navier-Stokes solutions, constructed particularly for an unstructured tetrahedral mesh. The scheme demonstrates robustness in high-speed flow computation and exhibits excellent adaptability to meshes with complex geometrical configurations.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Alsadig Ali, Abdullah Al-Mamun, Felipe Pereira, Arunasalam Rahunanthan
Summary: This paper presents a novel Bayesian statistical framework for the characterization of natural subsurface formations, and introduces the concept of multiscale sampling to localize the search in the stochastic space. The results show that the proposed framework performs well in solving inverse problems related to porous media flows.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Jacob Rains, Yi Wang, Alec House, Andrew L. Kaminsky, Nathan A. Tison, Vamshi M. Korivi
Summary: This paper presents a novel method called constrained optimized DMD with Control (cOptDMDc), which extends the optimized DMD method to systems with exogenous inputs and can enforce the stability of the resulting reduced order model (ROM). The proposed method optimally places eigenvalues within the stable region, thus mitigating spurious eigenvalue issues. Comparative studies show that cOptDMDc achieves high accuracy and robustness.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Andrea La Spina, Jacob Fish
Summary: This work introduces a hybridizable discontinuous Galerkin formulation for simulating ideal plasmas. The proposed method couples the fluid and electromagnetic subproblems monolithically based on source and employs a fully implicit time integration scheme. The approach also utilizes a projection-based divergence correction method to enforce the Gauss laws in challenging scenarios. Numerical examples demonstrate the high-order accuracy, efficiency, and robustness of the proposed formulation.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Junhong Yue, Peijun Li
Summary: This paper proposes two numerical methods (IP-FEM and BP-FEM) to study the flexural wave scattering problem of an arbitrary-shaped cavity on an infinite thin plate. These methods successfully decompose the fourth-order plate wave equation into the Helmholtz and modified Helmholtz equations with coupled conditions on the cavity boundary, providing an effective solution to this challenging problem.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
William Anderson, Mohammad Farazmand
Summary: We develop fast and scalable methods, called RONS, for computing reduced-order nonlinear solutions. These methods have been proven to be highly effective in tackling challenging problems, but become computationally prohibitive as the number of parameters grows. To address this issue, three separate methods are proposed and their efficacy is demonstrated through examples. The application of RONS to neural networks is also discussed.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Marco Caliari, Fabio Cassini
Summary: In this paper, a second order exponential scheme for stiff evolutionary advection-diffusion-reaction equations is proposed. The scheme is based on a directional splitting approach and uses computation of small sized exponential-like functions and tensor-matrix products for efficient implementation. Numerical examples demonstrate the advantage of the proposed approach over state-of-the-art techniques.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Sebastiano Boscarino, Seung Yeon Cho, Giovanni Russo
Summary: This work proposes a high order conservative semi-Lagrangian method for the inhomogeneous Boltzmann equation of rarefied gas dynamics. The method combines a semi-Lagrangian scheme for the convection term, a fast spectral method for computation of the collision operator, and a high order conservative reconstruction and a weighted optimization technique to preserve conservative quantities. Numerical tests demonstrate the accuracy and efficiency of the proposed method.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Jialei Li, Xiaodong Liu, Qingxiang Shi
Summary: This study shows that the number, centers, scattering strengths, inner and outer diameters of spherical shell-structured sources can be uniquely determined from the far field patterns. A numerical scheme is proposed for reconstructing the spherical shell-structured sources, which includes a migration series method for locating the centers and an iterative method for computing the inner and outer diameters without computing derivatives.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)