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Single-Step Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Method for 1-D Euler Equations Jayesh Badwaik, Praveen Chandrashekar, Christian Klingenberg Communications on Applied Mathematics and Computation    2020, 2 (4): 541-579.   DOI: 10.1007/s42967-019-00054-5 Abstract （15472）      PDF（pc） （5780KB）（1104）       Knowledge map    Save We propose an explicit, single-step discontinuous Galerkin method on moving grids using the arbitrary Lagrangian-Eulerian approach for one-dimensional Euler equations. The grid is moved with the local fuid velocity modifed by some smoothing, which is found to considerably reduce the numerical dissipation introduced by Riemann solvers. The scheme preserves constant states for any mesh motion and we also study its positivity preservation property. Local grid refnement and coarsening are performed to maintain the mesh quality and avoid the appearance of very small or large cells. Second, higher order methods are developed and several test cases are provided to demonstrate the accuracy of the proposed scheme. Reference | Related Articles | Metrics | Comments（0）

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Discrete Vector Calculus and Helmholtz Hodge Decomposition for Classical Finite Diference Summation by Parts Operators Hendrik Ranocha, Katharina Ostaszewski, Philip Heinisch Communications on Applied Mathematics and Computation    2020, 2 (4): 581-611.   DOI: 10.1007/s42967-019-00057-2 Abstract （662）      PDF（pc） （3566KB）（345）       Knowledge map    Save In this article, discrete variants of several results from vector calculus are studied for classical fnite diference summation by parts operators in two and three space dimensions. It is shown that existence theorems for scalar/vector potentials of irrotational/solenoidal vector felds cannot hold discretely because of grid oscillations, which are characterised explicitly. This results in a non-vanishing remainder associated with grid oscillations in the discrete Helmholtz Hodge decomposition. Nevertheless, iterative numerical methods based on an interpretation of the Helmholtz Hodge decomposition via orthogonal projections are proposed and applied successfully. In numerical experiments, the discrete remainder vanishes and the potentials converge with the same order of accuracy as usual in other frst-order partial diferential equations. Motivated by the successful application of the Helmholtz Hodge decomposition in theoretical plasma physics, applications to the discrete analysis of magnetohydrodynamic (MHD) wave modes are presented and discussed. Reference | Related Articles | Metrics | Comments（0）

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Nonlocal Flocking Dynamics: Learning the Fractional Order of PDEs from Particle Simulations Zhiping Mao, Zhen Li, George Em Karniadakis Communications on Applied Mathematics and Computation    2019, 1 (4): 597-619.   DOI: 10.1007/s42967-019-00031-y Abstract （8800）      PDF       Knowledge map    Save Flocking refers to collective behavior of a large number of interacting entities, where the interactions between discrete individuals produce collective motion on the large scale. We employ an agent-based model to describe the microscopic dynamics of each individual in a fock, and use a fractional partial diferential equation (fPDE) to model the evolution of macroscopic quantities of interest. The macroscopic models with phenomenological interaction functions are derived by applying the continuum hypothesis to the microscopic model. Instead of specifying the fPDEs with an ad hoc fractional order for nonlocal focking dynamics, we learn the efective nonlocal infuence function in fPDEs directly from particle trajectories generated by the agent-based simulations. We demonstrate how the learning framework is used to connect the discrete agent-based model to the continuum fPDEs in one- and two-dimensional nonlocal focking dynamics. In particular, a Cucker-Smale particle model is employed to describe the microscale dynamics of each individual, while Euler equations with nonlocal interaction terms are used to compute the evolution of macroscale quantities. The trajectories generated by the particle simulations mimic the feld data of tracking logs that can be obtained experimentally. They can be used to learn the fractional order of the infuence function using a Gaussian process regression model implemented with the Bayesian optimization. We show in one- and two-dimensional benchmarks that the numerical solution of the learned Euler equations solved by the fnite volume scheme can yield correct density distributions consistent with the collective behavior of the agent-based system solved by the particle method. The proposed method ofers new insights into how to scale the discrete agent-based models to the continuum-based PDE models, and could serve as a paradigm on extracting efective governing equations for nonlocal focking dynamics directly from particle trajectories. Reference | Related Articles | Metrics | Comments（0）

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Efficient Difference Schemes for the Caputo-Tempered Fractional Diffusion Equations Based on Polynomial Interpolation Le Zhao, Can Li, Fengqun Zhao Communications on Applied Mathematics and Computation    2021, 3 (1): 1-40.   DOI: 10.1007/s42967-020-00067-5 Abstract （1272）      PDF（pc） （4910KB）（302）       Knowledge map    Save The tempered fractional calculus has been successfully applied for depicting the time evolution of a system describing non-Markovian diffusion particles. The related governing equations are a series of partial differential equations with tempered fractional derivatives. Using the polynomial interpolation technique, in this paper, we present three efficient numerical formulas, namely the tempered L1 formula, the tempered L1-2 formula, and the tempered L2-1σ formula, to approximate the Caputo-tempered fractional derivative of order α∈(0,1). The truncation error of the tempered L1 formula is of order 2-α, and the tempered L1-2 formula and L2-1σ formula are of order 3-α. As an application, we construct implicit schemes and implicit ADI schemes for one-dimensional and two-dimensional time-tempered fractional diffusion equations, respectively. Furthermore, the unconditional stability and convergence of two developed difference schemes with tempered L1 and L2-1σ formulas are proved by the Fourier analysis method. Finally, we provide several numerical examples to demonstrate the correctness and effectiveness of the theoretical analysis. Reference | Related Articles | Metrics | Comments（0）

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A Finite Diference Method for Space Fractional Diferential Equations with Variable Difusivity Coefcient K. A. Mustapha, K. M. Furati, O. M. Knio, O. P. Le Maître Communications on Applied Mathematics and Computation    2020, 2 (4): 671-688.   DOI: 10.1007/s42967-020-00066-6 Abstract （788）      PDF（pc） （2321KB）（277）       Knowledge map    Save Anomalous difusion is a phenomenon that cannot be modeled accurately by second-order difusion equations, but is better described by fractional difusion models. The nonlocal nature of the fractional difusion operators makes substantially more difcult the mathematical analysis of these models and the establishment of suitable numerical schemes. This paper proposes and analyzes the frst fnite diference method for solving variable-coefcient onedimensional (steady state) fractional diferential equations (DEs) with two-sided fractional derivatives (FDs). The proposed scheme combines frst-order forward and backward Euler methods for approximating the left-sided FD when the right-sided FD is approximated by two consecutive applications of the frst-order backward Euler method. Our scheme reduces to the standard second-order central diference in the absence of FDs. The existence and uniqueness of the numerical solution are proved, and truncation errors of order h are demonstrated (h denotes the maximum space step size). The numerical tests illustrate the global O(h) accuracy, except for nonsmooth cases which, as expected, have deteriorated convergence rates. Reference | Related Articles | Metrics | Comments（0）

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Numerical Analysis of Linear and Nonlinear Time-Fractional Subdiffusion Equations Yubo Yang, Fanhai Zeng Communications on Applied Mathematics and Computation    2019, 1 (4): 621-637.   DOI: 10.1007/s42967-019-00033-w Abstract （16407）      PDF       Knowledge map    Save In this paper, a new type of the discrete fractional Grönwall inequality is developed, which is applied to analyze the stability and convergence of a Galerkin spectral method for a linear time-fractional subdiffusion equation. Based on the temporal-spatial error splitting argument technique, the discrete fractional Grönwall inequality is also applied to prove the unconditional convergence of a semi-implicit Galerkin spectral method for a nonlinear time-fractional subdiffusion equation. Reference | Related Articles | Metrics | Comments（0）

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Randomized Generalized Singular Value Decomposition Wei Wei, Hui Zhang, Xi Yang, Xiaoping Chen Communications on Applied Mathematics and Computation    2021, 3 (1): 137-156.   DOI: 10.1007/s42967-020-00061-x Abstract （1297）      PDF（pc） （2742KB）（271）       Knowledge map    Save The generalized singular value decomposition (GSVD) of two matrices with the same number of columns is a very useful tool in many practical applications. However, the GSVD may suffer from heavy computational time and memory requirement when the scale of the matrices is quite large. In this paper, we use random projections to capture the most of the action of the matrices and propose randomized algorithms for computing a low-rank approximation of the GSVD. Serval error bounds of the approximation are also presented for the proposed randomized algorithms. Finally, some experimental results show that the proposed randomized algorithms can achieve a good accuracy with less computational cost and storage requirement. Reference | Related Articles | Metrics | Comments（0）

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A Weak Galerkin Harmonic Finite Element Method for Laplace Equation Ahmed Al-Taweel, Yinlin Dong, Saqib Hussain, Xiaoshen Wang Communications on Applied Mathematics and Computation    2021, 3 (3): 527-544.   DOI: 10.1007/s42967-020-00097-z Abstract （2480）      PDF       Knowledge map    Save In this article, a weak Galerkin finite element method for the Laplace equation using the harmonic polynomial space is proposed and analyzed. The idea of using the ${P_k}$-harmonic polynomial space instead of the full polynomial space ${P_k}$ is to use a much smaller number of basis functions to achieve the same accuracy when k ≥ 2. The optimal rate of convergence is derived in both H1 and L2 norms. Numerical experiments have been conducted to verify the theoretical error estimates. In addition, numerical comparisons of using the P2-harmonic polynomial space and using the standard P2 polynomial space are presented. Reference | Related Articles | Metrics | Comments（0）

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A Note on Numerical Algorithm for the Time-Caputo and Space-Riesz Fractional Difusion Equation Junhong Tian, Hengfei Ding Communications on Applied Mathematics and Computation    2021, 3 (4): 571-584.   DOI: 10.1007/s42967-021-00139-0 Abstract （6192）      PDF       Knowledge map    Save Recently, Zhang and Ding developed a novel fnite diference scheme for the timeCaputo and space-Riesz fractional difusion equation with the convergence order $\mathcal{O}\left(\tau^{2-\alpha}+h^{2}\right)$ in Zhang and Ding (Commun. Appl. Math. Comput. 2(1): 57–72, 2020). Unfortunately, they only gave the stability and convergence results for α ∈ (0, 1) and $\beta \in\left[\frac{7}{8}+\frac{\sqrt[3]{621+48 \sqrt{87}}}{24}+\frac{19}{8 \sqrt[3]{621+48 \sqrt{87}}}, 2\right]$. In this paper, using a new analysis method, we fnd that the original diference scheme is unconditionally stable and convergent with order $\mathcal{O}\left(\tau^{2-\alpha}+h^{2}\right)$ for all α ∈ (0, 1) and β ∈ (1, 2]. Finally, some numerical examples are given to verify the correctness of the results. Reference | Related Articles | Metrics | Comments（0）

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A Note on the Adaptive Simpler Block GMRES Method Qiaohua Liu, Lei Yao, Aijing Liu Communications on Applied Mathematics and Computation    2019, 1 (3): 435-447.   DOI: 10.1007/s42967-019-00022-z Abstract （16046）      PDF       Knowledge map    Save The adaptive simpler block GMRES method was investigated by Zhong et al. (J Comput Appl Math 282:139-156, 2015) where the condition number of the adaptively chosen basis for the Krylov subspace was evaluated. In this paper, the new upper bound for the condition number is investigated. Numerical tests show that the new upper bound is tighter. Reference | Related Articles | Metrics | Comments（0）

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Learning the Mapping x ↦ $\sum\limits_{i = 1}^d {x_i^2}$: the Cost of Finding the Needle in a Haystack Jiefu Zhang, Leonardo Zepeda-Núñez, Yuan Yao, Lin Lin Communications on Applied Mathematics and Computation    2021, 3 (2): 313-335.   DOI: 10.1007/s42967-020-00078-2 Abstract （2762）      PDF       Knowledge map    Save The task of using the machine learning to approximate the mapping x ↦ $\sum\limits_{i = 1}^d {x_i^2}$ with xi ∈ [-1, 1] seems to be a trivial one. Given the knowledge of the separable structure of the function, one can design a sparse network to represent the function very accurately, or even exactly. When such structural information is not available, and we may only use a dense neural network, the optimization procedure to fnd the sparse network embedded in the dense network is similar to fnding the needle in a haystack, using a given number of samples of the function. We demonstrate that the cost (measured by sample complexity) of fnding the needle is directly related to the Barron norm of the function. While only a small number of samples are needed to train a sparse network, the dense network trained with the same number of samples exhibits large test loss and a large generalization gap. To control the size of the generalization gap, we fnd that the use of the explicit regularization becomes increasingly more important as d increases. The numerically observed sample complexity with explicit regularization scales as $\mathcal{O}$(d2.5), which is in fact better than the theoretically predicted sample complexity that scales as $\mathcal{O}$(d4). Without the explicit regularization (also called the implicit regularization), the numerically observed sample complexity is signifcantly higher and is close to $\mathcal{O}$(d4.5). Reference | Related Articles | Metrics | Comments（0）

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A Class of Preconditioners Based on Positive-Definite Operator Splitting Iteration Methods for Variable-Coefficient Space-Fractional Diffusion Equations Jun-Feng Yin, Yi-Shu Du Communications on Applied Mathematics and Computation    2021, 3 (1): 157-176.   DOI: 10.1007/s42967-020-00069-3 Abstract （1184）      PDF（pc） （3359KB）（220）       Knowledge map    Save After discretization by the finite volume method, the numerical solution of fractional diffusion equations leads to a linear system with the Toeplitz-like structure. The theoretical analysis gives sufficient conditions to guarantee the positive-definite property of the discretized matrix. Moreover, we develop a class of positive-definite operator splitting iteration methods for the numerical solution of fractional diffusion equations, which is unconditionally convergent for any positive constant. Meanwhile, the iteration methods introduce a new preconditioner for Krylov subspace methods. Numerical experiments verify the convergence of the positive-definite operator splitting iteration methods and show the efficiency of the proposed preconditioner, compared with the existing approaches. Reference | Related Articles | Metrics | Comments（0）

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A Unified Petrov–Galerkin Spectral Method and Fast Solver for Distributed-Order Partial Differential Equations Mehdi Samiee, Ehsan Kharazmi, Mark M. Meerschaert, Mohsen Zayernouri Communications on Applied Mathematics and Computation    2021, 3 (1): 61-90.   DOI: 10.1007/s42967-020-00070-w Abstract （1202）      PDF（pc） （4100KB）（211）       Knowledge map    Save Fractional calculus and fractional-order modeling provide effective tools for modeling and simulation of anomalous diffusion with power-law scalings. In complex multi-fractal anomalous transport phenomena, distributed-order partial differential equations appear as tractable mathematical models, where the underlying derivative orders are distributed over a range of values, hence taking into account a wide range of multi-physics from ultraslow-to-standard-to-superdiffusion/wave dynamics. We develop a unified, fast, and stable Petrov–Galerkin spectral method for such models by employing Jacobi poly-fractonomials and Legendre polynomials as temporal and spatial basis/test functions, respectively. By defining the proper underlying distributed Sobolev spaces and their equivalent norms, we rigorously prove the well-posedness of the weak formulation, and thereby, we carry out the corresponding stability and error analysis. We finally provide several numerical simulations to study the performance and convergence of proposed scheme. Reference | Related Articles | Metrics | Comments（0）

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Second-Order Finite Diference/Spectral Element Formulation for Solving the Fractional Advection-Difusion Equation Mostafa Abbaszadeh, Hanieh Amjadian Communications on Applied Mathematics and Computation    2020, 2 (4): 653-669.   DOI: 10.1007/s42967-020-00060-y Abstract （634）      PDF（pc） （1880KB）（205）       Knowledge map    Save The main aim of this paper is to analyze the numerical method based upon the spectral element technique for the numerical solution of the fractional advection-difusion equation. The time variable has been discretized by a second-order fnite diference procedure. The stability and the convergence of the semi-discrete formula have been proven. Then, the spatial variable of the main PDEs is approximated by the spectral element method. The convergence order of the fully discrete scheme is studied. The basis functions of the spectral element method are based upon a class of Legendre polynomials. The numerical experiments confrm the theoretical results. Reference | Related Articles | Metrics | Comments（0）

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A Modified Weak Galerkin Finite Element Method for the Biharmonic Equation on Polytopal Meshes Ming Cui, Xiu Ye, Shangyou Zhang Communications on Applied Mathematics and Computation    2021, 3 (1): 91-105.   DOI: 10.1007/s42967-020-00071-9 Abstract （1204）      PDF（pc） （3004KB）（203）       Knowledge map    Save A modified weak Galerkin (MWG) finite element method is developed for solving the biharmonic equation. This method uses the same finite element space as that of the discontinuous Galerkin method, the space of discontinuous polynomials on polytopal meshes. But its formulation is simple, symmetric, positive definite, and parameter independent, without any of six inter-element face-integral terms in the formulation of the discontinuous Galerkin method. Optimal order error estimates in a discrete H2 norm are established for the corresponding finite element solutions. Error estimates in the L2 norm are also derived with a sub-optimal order of convergence for the lowest-order element and an optimal order of convergence for all high-order of elements. The numerical results are presented to confirm the theory of convergence. Reference | Related Articles | Metrics | Comments（0）

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Nonlocal Dynamics for Non-Gaussian Systems Arising in Biophysical Modeling Xiaoli Chen, Jinqiao Duan Communications on Applied Mathematics and Computation    2020, 2 (2): 201-213.   DOI: 10.1007/s42967-019-00046-5 Abstract （1989）      PDF       Knowledge map    Save The aim of this article is to review our recent work on nonlocal dynamics of non-Gaussian systems arising in a gene regulatory network. We have used the mean exit time, escape probability and maximal likely trajectory to quantify dynamical behaviors of a stochastic diferential system with non-Gaussian α-stable Lévy motions, to examine how the nonGaussianity index and noise intensity afect the gene transcription processes. Reference | Related Articles | Metrics | Comments（0）

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T-Jordan Canonical Form and T-Drazin Inverse Based on the T-Product Yun Miao, Liqun Qi, Yimin Wei Communications on Applied Mathematics and Computation    2021, 3 (2): 201-220.   DOI: 10.1007/s42967-019-00055-4 Abstract （5998）      PDF       Knowledge map    Save In this paper, we investigate the tensor similarity and propose the T-Jordan canonical form and its properties. The concepts of the T-minimal polynomial and the T-characteristic polynomial are proposed. As a special case, we present properties when two tensors commute based on the tensor T-product. We prove that the Cayley-Hamilton theorem also holds for tensor cases. Then, we focus on the tensor decompositions: T-polar, T-LU, T-QR and T-Schur decompositions of tensors are obtained. When an F-square tensor is not invertible with the T-product, we study the T-group inverse and the T-Drazin inverse which can be viewed as the extension of matrix cases. The expressions of the T-group and T-Drazin inverses are given by the T-Jordan canonical form. The polynomial form of the T-Drazin inverse is also proposed. In the last part, we give the T-core-nilpotent decomposition and show that the T-index and T-Drazin inverses can be given by a limit process. Reference | Related Articles | Metrics | Comments（0）

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High-Order Local Discontinuous Galerkin Algorithm with Time Second-Order Schemes for the Two-Dimensional Nonlinear Fractional Difusion Equation Min Zhang, Yang Liu, Hong Li Communications on Applied Mathematics and Computation    2020, 2 (4): 613-640.   DOI: 10.1007/s42967-019-00058-1 Abstract （693）      PDF（pc） （2890KB）（194）       Knowledge map    Save In this article, some high-order local discontinuous Galerkin (LDG) schemes based on some second-order θ approximation formulas in time are presented to solve a two-dimensional nonlinear fractional difusion equation. The unconditional stability of the LDG scheme is proved, and an a priori error estimate with O(hk+1 + △t2) is derived, where k ≥ 0 denotes the index of the basis function. Extensive numerical results with Qk(k=0, 1, 2, 3) elements are provided to confrm our theoretical results, which also show that the secondorder convergence rate in time is not impacted by the changed parameter θ. Reference | Related Articles | Metrics | Comments（0）

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The Nonlinear Lopsided HSS-Like Modulus-Based Matrix Splitting Iteration Method for Linear Complementarity Problems with Positive-Definite Matrices Lu Jia, Xiang Wang, Xiao-Yong Xiao Communications on Applied Mathematics and Computation    2021, 3 (1): 109-122.   DOI: 10.1007/s42967-019-00038-5 Abstract （1244）      PDF（pc） （1718KB）（193）       Knowledge map    Save In this paper, by means of constructing the linear complementarity problems into the corresponding absolute value equation, we raise an iteration method, called as the nonlinear lopsided HSS-like modulus-based matrix splitting iteration method, for solving the linear complementarity problems whose coefficient matrix in Rn×n is large sparse and positive definite. From the convergence analysis, it is appreciable to see that the proposed method will converge to its accurate solution under appropriate conditions. Numerical examples demonstrate that the presented method precede to other methods in practical implementation. Reference | Related Articles | Metrics | Comments（0）

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On Convergence of MRQI and IMRQI Methods for Hermitian Eigenvalue Problems Fang Chen, Cun, Qiang Miao, Galina V. Muratova Communications on Applied Mathematics and Computation    2021, 3 (1): 189-197.   DOI: 10.1007/s42967-020-00079-1 Abstract （1205）      PDF（pc） （1426KB）（192）       Knowledge map    Save Bai et al. proposed the multistep Rayleigh quotient iteration (MRQI) as well as its inexact variant (IMRQI) in a recent work (Comput. Math. Appl. 77: 2396–2406, 2019). These methods can be used to effectively compute an eigenpair of a Hermitian matrix. The convergence theorems of these methods were established under two conditions imposed on the initial guesses for the target eigenvalue and eigenvector. In this paper, we show that these two conditions can be merged into a relaxed one, so the convergence conditions in these theorems can be weakened, and the resulting convergence theorems are applicable to a broad class of matrices. In addition, we give detailed discussions about the new convergence condition and the corresponding estimates of the convergence errors, leading to rigorous convergence theories for both the MRQI and the IMRQI. Reference | Related Articles | Metrics | Comments（0）