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Published in last 1 year |  In last 2 years |  In last 3 years |  All Please wait a minute... For Selected: Download Citations EndNote Ris BibTeX Toggle Thumbnails Select 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 （16416）      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） Select 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 （16052）      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） Select C1-Conforming Quadrilateral Spectral Element Method for Fourth-Order Equations Huiyuan Li, Weikun Shan, Zhimin Zhang Communications on Applied Mathematics and Computation    2019, 1 (3): 403-434.   DOI: 10.1007/s42967-019-00041-w Abstract （15728）      PDF       Knowledge map    Save This paper is devoted to Professor Benyu Guo's open question on the C1-conforming quadrilateral spectral element method for fourth-order equations which has been endeavored for years. Starting with generalized Jacobi polynomials on the reference square, we construct the C1-conforming basis functions using the bilinear mapping from the reference square onto each quadrilateral element which fall into three categories-interior modes, edge modes, and vertex modes. In contrast to the triangular element, compulsively compensatory requirements on the global C1-continuity should be imposed for edge and vertex mode basis functions such that their normal derivatives on each common edge are reduced from rational functions to polynomials, which depend on only parameters of the common edge. It is amazing that the C1-conforming basis functions on each quadrilateral element contain polynomials in primitive variables, the completeness is then guaranteed and further confirmed by the numerical results on the Petrov-Galerkin spectral method for the non-homogeneous boundary value problem of fourth-order equations on an arbitrary quadrilateral. Finally, a C1-conforming quadrilateral spectral element method is proposed for the biharmonic eigenvalue problem, and numerical experiments demonstrate the effectiveness and efficiency of our spectral element method. Reference | Related Articles | Metrics | Comments（0） Select 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 （15481）      PDF（pc） （5780KB）（1132）       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） Select 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 （14560）      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） Select A Non-intrusive Correction Algorithm for Classifcation Problems with Corrupted Data Jun Hou, Tong Qin, Kailiang Wu, Dongbin Xiu Communications on Applied Mathematics and Computation    2021, 3 (2): 337-356.   DOI: 10.1007/s42967-020-00084-4 Abstract （14182）      PDF       Knowledge map    Save A novel correction algorithm is proposed for multi-class classifcation problems with corrupted training data. The algorithm is non-intrusive, in the sense that it post-processes a trained classifcation model by adding a correction procedure to the model prediction. The correction procedure can be coupled with any approximators, such as logistic regression, neural networks of various architectures, etc. When the training dataset is sufciently large, we theoretically prove (in the limiting case) and numerically show that the corrected models deliver correct classifcation results as if there is no corruption in the training data. For datasets of fnite size, the corrected models produce signifcantly better recovery results, compared to the models without the correction algorithm. All of the theoretical fndings in the paper are verifed by our numerical examples. Reference | Related Articles | Metrics | Comments（0） Select A Two-Level Additive Schwarz Preconditioner for Local C0 Discontinuous Galerkin Methods of Kirchhof Plates Jianguo Huang, Xuehai Huang Communications on Applied Mathematics and Computation    2019, 1 (2): 167-185.   DOI: 10.1007/s42967-019-0003-1 Abstract （13306）      PDF       Knowledge map    Save A two-level additive Schwarz preconditioner based on the overlapping domain decomposition approach is proposed for the local C0 discontinuous Galerkin (LCDG) method of Kirchhof plates. Then with the help of an intergrid transfer operator and its error estimates, it is proved that the condition number is bounded by O(1 + (H4/δ4)), where H is the diameter of the subdomains and δ measures the overlap among subdomains. And for some special cases of small overlap, the estimate can be improved as O(1 + (H3/δ3)). At last, some numerical results are reported to demonstrate the high efciency of the two-level additive Schwarz preconditioner. Reference | Related Articles | Metrics | Comments（0） Select Modeling and Computing of Fractional Convection Equation Changpin Li, Qian Yi Communications on Applied Mathematics and Computation    2019, 1 (4): 565-595.   DOI: 10.1007/s42967-019-00019-8 Abstract （12018）      PDF       Knowledge map    Save In this paper, we derive the fractional convection (or advection) equations (FCEs) (or FAEs) to model anomalous convection processes. Through using a continuous time random walk (CTRW) with power-law jump length distributions, we formulate the FCEs depicted by Riesz derivatives with order in (0, 1). The numerical methods for fractional convection operators characterized by Riesz derivatives with order lying in (0, 1) are constructed too. Then the numerical approximations to FCEs are studied in detail. By adopting the implicit Crank-Nicolson method and the explicit Lax-Wendrof method in time, and the secondorder numerical method to the Riesz derivative in space, we, respectively, obtain the unconditionally stable numerical scheme and the conditionally stable numerical one for the FCE with second-order convergence both in time and in space. The accuracy and efciency of the derived methods are verifed by numerical tests. The transport performance characterized by the derived fractional convection equation is also displayed through numerical simulations. Reference | Related Articles | Metrics | Comments（0） Select Higher Order Collocation Methods for Nonlocal Problems and Their Asymptotic Compatibility Burak Aksoylu, Fatih Celiker, George A. Gazonas Communications on Applied Mathematics and Computation    2020, 2 (2): 261-303.   DOI: 10.1007/s42967-019-00051-8 Abstract （11547）      PDF       Knowledge map    Save We study the convergence and asymptotic compatibility of higher order collocation methods for nonlocal operators inspired by peridynamics, a nonlocal formulation of continuum mechanics. We prove that the methods are optimally convergent with respect to the polynomial degree of the approximation. A numerical method is said to be asymptotically compatible if the sequence of approximate solutions of the nonlocal problem converges to the solution of the corresponding local problem as the horizon and the grid sizes simultaneously approach zero. We carry out a calibration process via Taylor series expansions and a scaling of the nonlocal operator via a strain energy density argument to ensure that the resulting collocation methods are asymptotically compatible. We fnd that, for polynomial degrees greater than or equal to two, there exists a calibration constant independent of the horizon size and the grid size such that the resulting collocation methods for the nonlocal difusion are asymptotically compatible. We verify these fndings through extensive numerical experiments. Reference | Related Articles | Metrics | Comments（0） Select Hermite-Discontinuous Galerkin Overset Grid Methods for the Scalar Wave Equation Oleksii Beznosov, Daniel Appel? Communications on Applied Mathematics and Computation    2021, 3 (3): 391-418.   DOI: 10.1007/s42967-020-00075-5 Abstract （9468）      PDF       Knowledge map    Save We present high order accurate numerical methods for the wave equation that combines efficient Hermite methods with geometrically flexible discontinuous Galerkin methods by using overset grids. Near boundaries we use thin boundary fitted curvilinear grids and in the volume we use Cartesian grids so that the computational complexity of the solvers approaches a structured Cartesian Hermite method. Unlike many other overset methods we do not need to add artificial dissipation but we find that the built-in dissipation of the Hermite and discontinuous Galerkin methods is sufficient to maintain the stability. By numerical experiments we demonstrate the stability, accuracy, efficiency, and the applicability of the methods to forward and inverse problems. Reference | Related Articles | Metrics | Comments（0） Select Conservative Discontinuous Galerkin/Hermite Spectral Method for the Vlasov-Poisson System Francis Filbet, Tao Xiong Communications on Applied Mathematics and Computation    2022, 4 (1): 34-59.   DOI: 10.1007/s42967-020-00089-z Abstract （8966）      PDF       Knowledge map    Save We propose a class of conservative discontinuous Galerkin methods for the Vlasov-Poisson system written as a hyperbolic system using Hermite polynomials in the velocity variable. These schemes are designed to be systematically as accurate as one wants with provable conservation of mass and possibly total energy. Such properties in general are hard to achieve within other numerical method frameworks for simulating the Vlasov-Poisson system. The proposed scheme employs the discontinuous Galerkin discretization for both the Vlasov and the Poisson equations, resulting in a consistent description of the distribution function and the electric feld. Numerical simulations are performed to verify the order of the accuracy and conservation properties. Reference | Related Articles | Metrics | Comments（0） Select Multigrid Methods for Time-Fractional Evolution Equations: A Numerical Study Bangti Jin, Zhi Zhou Communications on Applied Mathematics and Computation    2020, 2 (2): 163-177.   DOI: 10.1007/s42967-019-00042-9 Abstract （8958）      PDF       Knowledge map    Save In this work, we develop an efcient iterative scheme for a class of nonlocal evolution models involving a Caputo fractional derivative of order α(0, 1) in time. The fully discrete scheme is obtained using the standard Galerkin method with conforming piecewise linear fnite elements in space and corrected high-order BDF convolution quadrature in time. At each time step, instead of solving the linear algebraic system exactly, we employ a multigrid iteration with a Gauss-Seidel smoother to approximate the solution efciently. Illustrative numerical results for nonsmooth problem data are presented to demonstrate the app Reference | Related Articles | Metrics | Comments（0） Select 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 （8816）      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） Select Preface to the Focused Issue in Honor of Professor Philip Roe on the Occasion of His 80th Birthday Rémi Abgrall, Jennifer K. Ryan, Chi, Wang Shu Communications on Applied Mathematics and Computation    2020, 2 (3): 319-320.   DOI: 10.1007/s42967-020-00064-8 Abstract （8665）      PDF       Knowledge map    Save Related Articles | Metrics | Comments（0） Select A New Spectral Method Using Nonstandard Singular Basis Functions for Time-Fractional Diferential Equations Wenjie Liu, Li-Lian Wang, Shuhuang Xiang Communications on Applied Mathematics and Computation    2019, 1 (2): 207-230.   DOI: 10.1007/s42967-019-00012-1 Abstract （8636）      PDF       Knowledge map    Save In this paper, we introduce new non-polynomial basis functions for spectral approximation of time-fractional partial diferential equations (PDEs). Diferent from many other approaches, the nonstandard singular basis functions are defned from some generalised Birkhof interpolation problems through explicit inversion of some prototypical fractional initial value problem (FIVP) with a smooth source term. As such, the singularity of the new basis can be tailored to that of the singular solutions to a class of time-fractional PDEs, leading to spectrally accurate approximation. It also provides the acceptable solution to more general singular problems. Reference | Related Articles | Metrics | Comments（0） Select Convergence to Steady-State Solutions of the New Type of High-Order Multi-resolution WENO Schemes: a Numerical Study Jun Zhu, Chi, Wang Shu Communications on Applied Mathematics and Computation    2020, 2 (3): 429-460.   DOI: 10.1007/s42967-019-00044-7 Abstract （8521）      PDF       Knowledge map    Save A new type of high-order multi-resolution weighted essentially non-oscillatory (WENO) schemes (Zhu and Shu in J Comput Phys, 375: 659–683, 2018) is applied to solve for steady-state problems on structured meshes. Since the classical WENO schemes (Jiang and Shu in J Comput Phys, 126: 202–228, 1996) might sufer from slight post-shock oscillations (which are responsible for the residue to hang at a truncation error level), this new type of high-order fnite-diference and fnite-volume multi-resolution WENO schemes is applied to control the slight post-shock oscillations and push the residue to settle down to machine zero in steady-state simulations. This new type of multi-resolution WENO schemes uses the same large stencils as that of the same order classical WENO schemes, could obtain ffth-order, seventh-order, and ninth-order in smooth regions, and could gradually degrade to frst-order so as to suppress spurious oscillations near strong discontinuities. The linear weights of such new multi-resolution WENO schemes can be any positive numbers on the condition that their sum is one. This is the frst time that a series of unequal-sized hierarchical central spatial stencils are used in designing high-order fnitediference and fnite-volume WENO schemes for solving steady-state problems. In comparison with the classical ffth-order fnite-diference and fnite-volume WENO schemes, the residue of these new high-order multi-resolution WENO schemes can converge to a tiny number close to machine zero for some benchmark steady-state problems. Reference | Related Articles | Metrics | Comments（0） Select CAMC Focused Section on Tensor Computation Delin Chu, Michael Ng, Liqun Qi, Qing-Wen Wang Communications on Applied Mathematics and Computation    2021, 3 (2): 199-199.   DOI: 10.1007/s42967-020-00113-2 Abstract （8015）      PDF       Knowledge map    Save Related Articles | Metrics | Comments（0） Select An Efcient Second-Order Convergent Scheme for One-Side Space Fractional Difusion Equations with Variable Coefcients Xue-lei Lin, Pin Lyu, Michael K. Ng, Hai-Wei Sun, Seakweng Vong Communications on Applied Mathematics and Computation    2020, 2 (2): 215-239.   DOI: 10.1007/s42967-019-00050-9 Abstract （7839）      PDF       Knowledge map    Save In this paper, a second-order fnite-diference scheme is investigated for time-dependent space fractional difusion equations with variable coefcients. In the presented scheme, the Crank-Nicolson temporal discretization and a second-order weighted-and-shifted Grünwald-Letnikov spatial discretization are employed. Theoretically, the unconditional stability and the second-order convergence in time and space of the proposed scheme are established under some conditions on the variable coefcients. Moreover, a Toeplitz preconditioner is proposed for linear systems arising from the proposed scheme. The condition number of the preconditioned matrix is proven to be bounded by a constant independent of the discretization step-sizes, so that the Krylov subspace solver for the preconditioned linear systems converges linearly. Numerical results are reported to show the convergence rate and the efciency of the proposed scheme. Reference | Related Articles | Metrics | Comments（0） Select Parallel Active Subspace Decomposition for Tensor Robust Principal Component Analysis Michael K. Ng, Xue-Zhong Wang Communications on Applied Mathematics and Computation    2021, 3 (2): 221-241.   DOI: 10.1007/s42967-020-00063-9 Abstract （7013）      PDF       Knowledge map    Save Tensor robust principal component analysis has received a substantial amount of attention in various felds. Most existing methods, normally relying on tensor nuclear norm minimization, need to pay an expensive computational cost due to multiple singular value decompositions at each iteration. To overcome the drawback, we propose a scalable and efcient method, named parallel active subspace decomposition, which divides the unfolding along each mode of the tensor into a columnwise orthonormal matrix (active subspace) and another small-size matrix in parallel. Such a transformation leads to a nonconvex optimization problem in which the scale of nuclear norm minimization is generally much smaller than that in the original problem. We solve the optimization problem by an alternating direction method of multipliers and show that the iterates can be convergent within the given stopping criterion and the convergent solution is close to the global optimum solution within the prescribed bound. Experimental results are given to demonstrate that the performance of the proposed model is better than the state-of-the-art methods. Reference | Related Articles | Metrics | Comments（0） Select Preface to Focused Section on Efcient HighcOrder Time Discretization Methods for Partial Diferential Equations Sebastiano Boscarino, Giuseppe Izzo, Lorenzo Pareschi, Giovanni Russo, Chi-Wang Shu Communications on Applied Mathematics and Computation    2021, 3 (4): 605-605.   DOI: 10.1007/s42967-021-00164-z Abstract （6846）      PDF       Knowledge map    Save Related Articles | Metrics | Comments（0）