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On the Complexity of Finding Tensor Ranks Mohsen Aliabadi, Shmuel Friedland Communications on Applied Mathematics and Computation    2021, 3 (2): 281-289.   DOI: 10.1007/s42967-020-00103-4 Abstract （2333）      PDF       Knowledge map    Save The purpose of this note is to give a linear algebra algorithm to fnd out if a rank of a given tensor over a feld $\mathbb{F}$ is at most k over the algebraic closure of $\mathbb{F}$, where k is a given positive integer. We estimate the arithmetic complexity of our algorithm. Reference | Related Articles | Metrics | Comments（0）

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An Order Reduction Method for the Nonlinear Caputo-Hadamard Fractional Diffusion-Wave Model Jieying Zhang, Caixia Ou, Zhibo Wang, Seakweng Vong Communications on Applied Mathematics and Computation    2025, 7 (1): 392-408.   DOI: 10.1007/s42967-023-00295-5 Abstract （12）      PDF       Knowledge map    Save In this paper, the numerical solutions of the nonlinear Hadamard fractional diffusion-wave model with the initial singularity are investigated. Firstly, the model is transformed into coupled equations by virtue of a symmetric fractional-order reduction method. Then the Llog,2-1σ formula on nonuniform grids is applied to approach to the time fractional derivative. In addition, the discrete fractional Grönwall inequality is used to analyze the optimal convergence of the constructed numerical scheme by the energy method. The accuracy of the theoretical analysis will be demonstrated by means of a numerical experiment at the end. 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 （1016）      PDF（pc） （2321KB）（500）       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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Mathematical Modeling of Biological Fluid Flow Through a Cylindrical Layer with Due Account for Barodiffusion N. N. Nazarenko, A. G. Knyazeva Communications on Applied Mathematics and Computation    2023, 5 (4): 1365-1384.   DOI: 10.1007/s42967-022-00203-3 Abstract （176）      PDF       Knowledge map    Save The work proposes a model of biological fluid flow in a steady mode through a cylindrical layer taking into account convection and diffusion. The model considers finite compressibility and concentration expansion connected with both barodiffusion and additional mechanism of pressure change in the pore volume due to the concentration gradient. Thus, the model is entirely coupled. The paper highlights the complexes composed of scales of physical quantities of different natures. The iteration algorithm for the numerical solution of the problem was developed for the coupled problem. The work involves numerical studies of the considered effects on the characteristics of the flow that can be convective or diffusive, depending on the relation between the dimensionless complexes. It is demonstrated that the distribution of velocity and concentration for different cylinder wall thicknesses is different. It is established that the barodiffusion has a considerable impact on the process in the convective mode or in the case of reduced cylinder wall thickness. Reference | Related Articles | Metrics | Comments（0）

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Dynamics of Advantageous Mutant Spread in Spatial Death-Birth and Birth-Death Moran Models Jasmine Foo, Einar Bjarki Gunnarsson, Kevin Leder, David Sivakoff Communications on Applied Mathematics and Computation    2024, 6 (1): 576-604.   DOI: 10.1007/s42967-023-00278-6 Abstract （130）      PDF       Knowledge map    Save The spread of an advantageous mutation through a population is of fundamental interest in population genetics. While the classical Moran model is formulated for a well-mixed population, it has long been recognized that in real-world applications, the population usually has an explicit spatial structure which can significantly influence the dynamics. In the context of cancer initiation in epithelial tissue, several recent works have analyzed the dynamics of advantageous mutant spread on integer lattices, using the biased voter model from particle systems theory. In this spatial version of the Moran model, individuals first reproduce according to their fitness and then replace a neighboring individual. From a biological standpoint, the opposite dynamics, where individuals first die and are then replaced by a neighboring individual according to its fitness, are equally relevant. Here, we investigate this death-birth analogue of the biased voter model. We construct the process mathematically, derive the associated dual process, establish bounds on the survival probability of a single mutant, and prove that the process has an asymptotic shape. We also briefly discuss alternative birth-death and death-birth dynamics, depending on how the mutant fitness advantage affects the dynamics. We show that birth-death and death-birth formulations of the biased voter model are equivalent when fitness affects the former event of each update of the model, whereas the birth-death model is fundamentally different from the death-birth model when fitness affects the latter event. Reference | Related Articles | Metrics | Comments（0）

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A Semi-Lagrangian Spectral Method for the Vlasov-Poisson System Based on Fourier, Legendre and Hermite Polynomials Lorella Fatone, Daniele Funaro, Gianmarco Manzini Communications on Applied Mathematics and Computation    2019, 1 (3): 333-360.   DOI: 10.1007/s42967-019-00027-8 Abstract （719）      PDF       Knowledge map    Save In this work, we apply a semi-Lagrangian spectral method for the Vlasov-Poisson system, previously designed for periodic Fourier discretizations, by implementing Legendre polynomials and Hermite functions in the approximation of the distribution function with respect to the velocity variable. We discuss second-order accurate-in-time schemes, obtained by coupling spectral techniques in the space-velocity domain with a BDF timestepping scheme. The resulting method possesses good conservation properties, which have been assessed by a series of numerical tests conducted on some standard benchmark problems including the two-stream instability and the Landau damping test cases. In the Hermite case, we also investigate the numerical behavior in dependence of a scaling parameter in the Gaussian weight. Confirming previous results from the literature, our experiments for different representative values of this parameter, indicate that a proper choice may significantly impact on accuracy, thus suggesting that suitable strategies should be developed to automatically update the parameter during the time-advancing procedure. Reference | Related Articles | Metrics | Comments（0）

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Numerical Investigations on the Resonance Errors of Multiscale Discontinuous Galerkin Methods for One-Dimensional Stationary Schrödinger Equation Bo Dong, Wei Wang Communications on Applied Mathematics and Computation    2024, 6 (1): 311-324.   DOI: 10.1007/s42967-022-00248-4 Abstract （151）      PDF       Knowledge map    Save In this paper, numerical experiments are carried out to investigate the impact of penalty parameters in the numerical traces on the resonance errors of high-order multiscale discontinuous Galerkin (DG) methods (Dong et al. in J Sci Comput 66: 321-345, 2016; Dong and Wang in J Comput Appl Math 380: 1-11, 2020) for a one-dimensional stationary Schrödinger equation. Previous work showed that penalty parameters were required to be positive in error analysis, but the methods with zero penalty parameters worked fine in numerical simulations on coarse meshes. In this work, by performing extensive numerical experiments, we discover that zero penalty parameters lead to resonance errors in the multiscale DG methods, and taking positive penalty parameters can effectively reduce resonance errors and make the matrix in the global linear system have better condition numbers. Reference | Related Articles | Metrics | Comments（0）

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Superconvergence of Energy-Conserving Discontinuous Galerkin Methods for Linear Hyperbolic Equations Yong Liu, Chi-Wang Shu, Mengping Zhang Communications on Applied Mathematics and Computation    2019, 1 (1): 101-116.   DOI: 10.1007/s42967-019-0006-y Abstract （1027）      PDF       Knowledge map    Save In this paper, we study the superconvergence properties of the energy-conserving discontinuous Galerkin (DG) method in[18] for one-dimensional linear hyperbolic equations. We prove the approximate solution superconverges to a particular projection of the exact solution. The order of this superconvergence is proved to be k + 2 when piecewise Pk polynomials with k ≥ 1 are used. The proof is valid for arbitrary non-uniform regular meshes and for piecewise Pk polynomials with arbitrary k ≥ 1. Furthermore, we find that the derivative and function value approximations of the DG solution are superconvergent at a class of special points, with an order of k + 1 and k + 2, respectively. We also prove, under suitable choice of initial discretization, a (2k + 1)-th order superconvergence rate of the DG solution for the numerical fluxes and the cell averages. Numerical experiments are given to demonstrate these theoretical results. Reference | Related Articles | Metrics | Comments（0）

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A Local Macroscopic Conservative (LoMaC) Low Rank Tensor Method with the Discontinuous Galerkin Method for the Vlasov Dynamics Wei Guo, Jannatul Ferdous Ema, Jing-Mei Qiu Communications on Applied Mathematics and Computation    2024, 6 (1): 550-575.   DOI: 10.1007/s42967-023-00277-7 Abstract （123）      PDF       Knowledge map    Save In this paper, we propose a novel Local Macroscopic Conservative (LoMaC) low rank tensor method with discontinuous Galerkin (DG) discretization for the physical and phase spaces for simulating the Vlasov-Poisson (VP) system. The LoMaC property refers to the exact local conservation of macroscopic mass, momentum, and energy at the discrete level. The recently developed LoMaC low rank tensor algorithm (arXiv: 2207.00518) simultaneously evolves the macroscopic conservation laws of mass, momentum, and energy using the kinetic flux vector splitting; then the LoMaC property is realized by projecting the low rank kinetic solution onto a subspace that shares the same macroscopic observables. This paper is a generalization of our previous work, but with DG discretization to take advantage of its compactness and flexibility in handling boundary conditions and its superior accuracy in the long term. The algorithm is developed in a similar fashion as that for a finite difference scheme, by observing that the DG method can be viewed equivalently in a nodal fashion. With the nodal DG method, assuming a tensorized computational grid, one will be able to (i) derive differentiation matrices for different nodal points based on a DG upwind discretization of transport terms, and (ii) define a weighted inner product space based on the nodal DG grid points. The algorithm can be extended to the high dimensional problems by hierarchical Tucker (HT) decomposition of solution tensors and a corresponding conservative projection algorithm. In a similar spirit, the algorithm can be extended to DG methods on nodal points of an unstructured mesh, or to other types of discretization, e.g., the spectral method in velocity direction. Extensive numerical results are performed to showcase the efficacy of the method. Reference | Related Articles | Metrics | Comments（0）

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Semi-implicit-Type Order-Adaptive CAT2 Schemes for Systems of Balance Laws with Relaxed Source Term Emanuele Macca, Sebastiano Boscarino Communications on Applied Mathematics and Computation    2025, 7 (1): 151-178.   DOI: 10.1007/s42967-024-00414-w Abstract （10）      PDF       Knowledge map    Save In this paper, we present two semi-implicit-type second-order compact approximate Taylor (CAT2) numerical schemes and blend them with a local a posteriori multi-dimensional optimal order detection (MOOD) paradigm to solve hyperbolic systems of balance laws with relaxed source terms. The resulting scheme presents the high accuracy when applied to smooth solutions, essentially non-oscillatory behavior for irregular ones, and offers a nearly fail-safe property in terms of ensuring the positivity. The numerical results obtained from a variety of test cases, including smooth and non-smooth well-prepared and unprepared initial conditions, assessing the appropriate behavior of the semi-implicit-type second order CATMOOD schemes. These results have been compared in the accuracy and the efficiency with a second-order semi-implicit Runge-Kutta (RK) method. Reference | Related Articles | Metrics | Comments（0）

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A Numerical Algorithm for the Caputo Tempered Fractional Advection-Diffusion Equation Wenhui Guan, Xuenian Cao Communications on Applied Mathematics and Computation    2021, 3 (1): 41-59.   DOI: 10.1007/s42967-020-00080-8 Abstract （1826）      PDF（pc） （2238KB）（377）       Knowledge map    Save By transforming the Caputo tempered fractional advection-diffusion equation into the Riemann–Liouville tempered fractional advection-diffusion equation, and then using the fractional-compact Grünwald–Letnikov tempered difference operator to approximate the Riemann–Liouville tempered fractional partial derivative, the fractional central difference operator to discritize the space Riesz fractional partial derivative, and the classical central difference formula to discretize the advection term, a numerical algorithm is constructed for solving the Caputo tempered fractional advection-diffusion equation. The stability and the convergence analysis of the numerical method are given. Numerical experiments show that the numerical method is effective. Reference | Related Articles | Metrics | Comments（0）

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A Provable Positivity-Preserving Local Discontinuous Galerkin Method for the Viscous and Resistive MHD Equations Mengjiao Jiao, Yan Jiang, Mengping Zhang Communications on Applied Mathematics and Computation    2024, 6 (1): 279-310.   DOI: 10.1007/s42967-022-00247-5 Abstract （101）      PDF       Knowledge map    Save In this paper, we construct a high-order discontinuous Galerkin (DG) method which can preserve the positivity of the density and the pressure for the viscous and resistive magnetohydrodynamics (VRMHD). To control the divergence error in the magnetic field, both the local divergence-free basis and the Godunov source term would be employed for the multi-dimensional VRMHD. Rigorous theoretical analyses are presented for one-dimensional and multi-dimensional DG schemes, respectively, showing that the scheme can maintain the positivity-preserving (PP) property under some CFL conditions when combined with the strong-stability-preserving time discretization. Then, general frameworks are established to construct the PP limiter for arbitrary order of accuracy DG schemes. Numerical tests demonstrate the effectiveness of the proposed schemes. Reference | Related Articles | Metrics | Comments（0）

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Local Discontinuous Galerkin Scheme for Space Fractional Allen-Cahn Equation Can Li, Shuming Liu Communications on Applied Mathematics and Computation    2020, 2 (1): 73-91.   DOI: 10.1007/s42967-019-00034-9 Abstract （14807）      PDF       Knowledge map    Save This paper is concerned with the efcient numerical solution for a space fractional Allen-Cahn (AC) equation. Based on the features of the fractional derivative, we design and analyze a semi-discrete local discontinuous Galerkin (LDG) scheme for the initial-boundary problem of the space fractional AC equation. We prove the optimal convergence rates of the semi-discrete LDG approximation for smooth solutions. Finally, we test the accuracy and efciency of the designed numerical scheme on a uniform grid by three examples. Numerical simulations show that the space fractional AC equation displays abundant dynamical behaviors. Reference | Related Articles | Metrics | Comments（0）

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Finite Difference Schemes for Time-Space Fractional Diffusion Equations in One- and Two-Dimensions Yu Wang, Min Cai Communications on Applied Mathematics and Computation    2023, 5 (4): 1674-1696.   DOI: 10.1007/s42967-022-00244-8 Abstract （310）      PDF       Knowledge map    Save In this paper, finite difference schemes for solving time-space fractional diffusion equations in one dimension and two dimensions are proposed. The temporal derivative is in the Caputo-Hadamard sense for both cases. The spatial derivative for the one-dimensional equation is of Riesz definition and the two-dimensional spatial derivative is given by the fractional Laplacian. The schemes are proved to be unconditionally stable and convergent. The numerical results are in line with the theoretical analysis. Reference | Related Articles | Metrics | Comments（0）

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Oscillation of Second-Order Half-Linear Neutral Advanced Differential Equations Shan Shi, Zhenlai Han Communications on Applied Mathematics and Computation    2021, 3 (3): 497-508.   DOI: 10.1007/s42967-020-00092-4 Abstract （1632）      PDF       Knowledge map    Save The purpose of this paper is to study the oscillation of second-order half-linear neutral differential equations with advanced argument of the form (r(t)((y(t) + p(t)y(τ(t)))')α)' + q(t)yα(σ(t))=0, t ≥ t0, when ∫∞ ${r^{-\frac{1}{\alpha }}}$(s)ds < ∞. We obtain sufficient conditions for the oscillation of the studied equations by the inequality principle and the Riccati transformation. An example is provided to illustrate the results. Reference | Related Articles | Metrics | Comments（0）

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Standard Dual Quaternion Optimization and Its Applications in Hand-Eye Calibration and SLAM Liqun Qi Communications on Applied Mathematics and Computation    2023, 5 (4): 1469-1483.   DOI: 10.1007/s42967-022-00213-1 Abstract （212）      PDF       Knowledge map    Save Several common dual quaternion functions, such as the power function, the magnitude function, the 2-norm function, and the kth largest eigenvalue of a dual quaternion Hermitian matrix, are standard dual quaternion functions, i.e., the standard parts of their function values depend upon only the standard parts of their dual quaternion variables. Furthermore, the sum, product, minimum, maximum, and composite functions of two standard dual functions, the logarithm and the exponential of standard unit dual quaternion functions, are still standard dual quaternion functions. On the other hand, the dual quaternion optimization problem, where objective and constraint function values are dual numbers but variables are dual quaternions, naturally arises from applications. We show that to solve an equality constrained dual quaternion optimization (EQDQO) problem, we only need to solve two quaternion optimization problems. If the involved dual quaternion functions are all standard, the optimization problem is called a standard dual quaternion optimization problem, and some better results hold. Then, we show that the dual quaternion optimization problems arising from the hand-eye calibration problem and the simultaneous localization and mapping (SLAM) problem are equality constrained standard dual quaternion optimization problems. Reference | Related Articles | Metrics | Comments（0）

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Superconvergence and the Numerical Flux: a Study Using the Upwind-Biased Flux in Discontinuous Galerkin Methods Daniel J. Frean, Jennifer K. Ryan Communications on Applied Mathematics and Computation    2020, 2 (3): 461-486.   DOI: 10.1007/s42967-019-00049-2 Abstract （954）      PDF       Knowledge map    Save One of the benefcial properties of the discontinuous Galerkin method is the accurate wave propagation properties. That is, the semi-discrete error has dissipation errors of order 2k + 1 (≤ Ch2k+1) and order 2k + 2 for dispersion (≤ Ch2k+2). Previous studies have concentrated on the order of accuracy, and neglected the important role that the error constant, C, plays in these estimates. In this article, we show the important role of the error constant in the dispersion and dissipation error for discontinuous Galerkin approximation of polynomial degree k, where k = 0, 1, 2, 3. This gives insight into why one may want a more centred fux for a piecewise constant or quadratic approximation than for a piecewise linear or cubic approximation. We provide an explicit formula for these error constants. This is illustrated through one particular fux, the upwind-biased fux introduced by Meng et al., as it is a convex combination of the upwind and downwind fuxes. The studies of wave propagation are typically done through a Fourier ansatz. This higher order Fourier information can be extracted using the smoothness-increasing accuracy-conserving (SIAC) flter. The SIAC flter ties the higher order Fourier information to the negative-order norm in physical space. We show that both the proofs of the ability of the SIAC flter to extract extra accuracy and numerical results are unafected by the choice of fux. Reference | Related Articles | Metrics | Comments（0）

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Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics Hendrik Ranocha, Lisandro Dalcin, Matteo Parsani, David I. Ketcheson Communications on Applied Mathematics and Computation    2022, 4 (4): 1191-1228.   DOI: 10.1007/s42967-021-00159-w Abstract （3217）      PDF       Knowledge map    Save We develop error-control based time integration algorithms for compressible fluid dynamics (CFD) applications and show that they are efficient and robust in both the accuracy-limited and stability-limited regime. Focusing on discontinuous spectral element semidiscretizations, we design new controllers for existing methods and for some new embedded Runge-Kutta pairs. We demonstrate the importance of choosing adequate controller parameters and provide a means to obtain these in practice. We compare a wide range of error-control-based methods, along with the common approach in which step size control is based on the Courant-Friedrichs-Lewy (CFL) number. The optimized methods give improved performance and naturally adopt a step size close to the maximum stable CFL number at loose tolerances, while additionally providing control of the temporal error at tighter tolerances. The numerical examples include challenging industrial CFD applications. 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 （1597）      PDF（pc） （2742KB）（460）       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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Adaptive Sparse Grid Discontinuous Galerkin Method: Review and Software Implementation Juntao Huang, Wei Guo, Yingda Cheng Communications on Applied Mathematics and Computation    2024, 6 (1): 501-532.   DOI: 10.1007/s42967-023-00268-8 Abstract （142）      PDF       Knowledge map    Save This paper reviews the adaptive sparse grid discontinuous Galerkin (aSG-DG) method for computing high dimensional partial differential equations (PDEs) and its software implementation. The C++ software package called AdaM-DG, implementing the aSG-DG method, is available on GitHub at https://github.com/JuntaoHuang/adaptive-multiresolution-DG. The package is capable of treating a large class of high dimensional linear and nonlinear PDEs. We review the essential components of the algorithm and the functionality of the software, including the multiwavelets used, assembling of bilinear operators, fast matrix-vector product for data with hierarchical structures. We further demonstrate the performance of the package by reporting the numerical error and the CPU cost for several benchmark tests, including linear transport equations, wave equations, and Hamilton-Jacobi (HJ) equations. Reference | Related Articles | Metrics | Comments（0）