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Conforming P3 Divergence-Free Finite Elements for the Stokes Equations on Subquadrilateral Triangular Meshes Shangyou Zhang Communications on Applied Mathematics and Computation    2025, 7 (2): 426-441.   DOI: 10.1007/s42967-023-00335-0 Abstract （9）      PDF       Knowledge map    Save The continuous P3 and discontinuous P2 finite element pair is stable on subquadrilateral triangular meshes for solving 2D stationary Stokes equations. By putting two diagonal lines into every quadrilateral of a quadrilateral mesh, we get a subquadrilateral triangular mesh. Such a velocity solution is divergence-free point wise and viscosity robust in the sense the solution and the error are independent of the viscosity. Numerical examples show an advantage of such a method over the Taylor-Hood P3-P2 method, where the latter deteriorates when the viscosity becomes small. Reference | Related Articles | Metrics | Comments（0）

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Preface to Focused Section on Efficient High-Order Time Discretization Methods for Partial Differential Equations Sebastiano Boscarino, Giuseppe Izzo, Lorenzo Pareschi, Giovanni Russo, Chi-Wang Shu Communications on Applied Mathematics and Computation    2025, 7 (1): 1-2.   DOI: 10.1007/s42967-024-00464-0 Abstract （10）      PDF       Knowledge map    Save Related Articles | Metrics | Comments（0）

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Preface Zhong-Zhi Bai, Wei Cai, Qiang Du, Weinan E, Chi-Wang Shu, Xuejun Xu, Zhimin Zhang Communications on Applied Mathematics and Computation    2025, 7 (2): 409-410.   DOI: 10.1007/s42967-024-00471-1 Abstract （11）      PDF       Knowledge map    Save 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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L1/LDG Method for Caputo-Hadamard Time Fractional Diffusion Equation Zhen Wang Communications on Applied Mathematics and Computation    2025, 7 (1): 203-227.   DOI: 10.1007/s42967-023-00257-x Abstract （20）      PDF       Knowledge map    Save In this paper, a class of discrete Gronwall inequalities is proposed. It is efficiently applied to analyzing the constructed L1/local discontinuous Galerkin (LDG) finite element methods which are used for numerically solving the Caputo-Hadamard time fractional diffusion equation. The derived numerical methods are shown to be α-robust using the newly established Gronwall inequalities, that is, it remains valid when α → 1-. Numerical experiments are given to demonstrate the theoretical statements. Reference | Related Articles | Metrics | Comments（0）

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On the Regularity of Time-Harmonic Maxwell Equations with Impedance Boundary Conditions Zhiming Chen Communications on Applied Mathematics and Computation    2025, 7 (2): 759-770.   DOI: 10.1007/s42967-024-00386-x Abstract （10）      PDF       Knowledge map    Save In this paper, we prove the H2 regularity of the solution to the time-harmonic Maxwell equations with impedance boundary conditions on domains with a C2 boundary under minimum regularity assumptions on the source and boundary functions. Reference | Related Articles | Metrics | Comments（0）

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The High-Order Variable-Coefficient Explicit-Implicit-Null Method for Diffusion and Dispersion Equations Meiqi Tan, Juan Cheng, Chi-Wang Shu Communications on Applied Mathematics and Computation    2025, 7 (1): 115-150.   DOI: 10.1007/s42967-023-00359-6 Abstract （29）      PDF       Knowledge map    Save For the high-order diffusion and dispersion equations, the general practice of the explicitimplicit-null (EIN) method is to add and subtract an appropriately large linear highest derivative term with a constant coefficient at one side of the equation, and then apply the standard implicit-explicit method to the equivalent equation. We call this approach the constant-coefficient EIN method in this paper and hereafter denote it by “CC-EIN”. To reduce the error in the CC-EIN method, the variable-coefficient explicit-implicit-null (VC-EIN) method, which is obtained by adding and subtracting a linear highest derivative term with a variable coefficient, is proposed and studied in this paper. Coupled with the local discontinuous Galerkin (LDG) spatial discretization, the VC-EIN method is shown to be unconditionally stable and can achieve high order of accuracy for both one-dimensional and twodimensional quasi-linear and nonlinear equations. In addition, although the computational cost slightly increases, the VC-EIN method can obtain more accurate results than the CCEIN method, if the diffusion coefficient or the dispersion coefficient has a few high and narrow bumps and the bumps only account for a small part of the whole computational domain. Reference | Related Articles | Metrics | Comments（0）

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Beyond Strang: a Practical Assessment of Some Second-Order 3-Splitting Methods Raymond J. Spiteri, Arash Tavassoli, Siqi Wei, Andrei Smolyakov Communications on Applied Mathematics and Computation    2025, 7 (1): 95-114.   DOI: 10.1007/s42967-023-00314-5 Abstract （6）      PDF       Knowledge map    Save Operator-splitting is a popular divide-and-conquer strategy for solving differential equations. Typically, the right-hand side of the differential equation is split into a number of parts that are then integrated separately. Many methods are known that split the righthand side into two parts. This approach is limiting, however, and there are situations when 3-splitting is more natural and ultimately more advantageous. The second-order Strang operator-splitting method readily generalizes to a right-hand side splitting into any number of operators. It is arguably the most popular method for 3-splitting because of its efficiency, ease of implementation, and intuitive nature. Other 3-splitting methods exist, but they are less well known, and analysis and evaluation of their performance in practice are scarce. We demonstrate the effectiveness of some alternative 3-splitting, second-order methods to Strang splitting on two problems: the reaction-diffusion Brusselator, which can be split into three parts that each have closed-form solutions, and the kinetic Vlasov-Poisson equations that are used in semi-Lagrangian plasma simulations. We find alternative second-order 3-operator-splitting methods that realize efficiency gains of 10%–20% over traditional Strang splitting. Our analysis for the practical assessment of the efficiency of operator-splitting methods includes the computational cost of the integrators and can be used in method design. Reference | Related Articles | Metrics | Comments（0）

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Asymmetry and Condition Number of an Elliptic-Parabolic System for Biological Network Formation Clarissa Astuto, Daniele Boffi, Jan Haskovec, Peter Markowich, Giovanni Russo Communications on Applied Mathematics and Computation    2025, 7 (1): 78-94.   DOI: 10.1007/s42967-023-00297-3 Abstract （18）      PDF       Knowledge map    Save We present results of numerical simulations of the tensor-valued elliptic-parabolic PDE model for biological network formation. The numerical method is based on a nonlinear finite difference scheme on a uniform Cartesian grid in a two-dimensional (2D) domain. The focus is on the impact of different discretization methods and choices of regularization parameters on the symmetry of the numerical solution. In particular, we show that using the symmetric alternating direction implicit (ADI) method for time discretization helps preserve the symmetry of the solution, compared to the (non-symmetric) ADI method. Moreover, we study the effect of the regularization by the isotropic background permeability r > 0, showing that the increased condition number of the elliptic problem due to decreasing value of r leads to loss of symmetry. We show that in this case, neither the use of the symmetric ADI method preserves the symmetry of the solution. Finally, we perform the numerical error analysis of our method making use of the Wasserstein distance. Reference | Related Articles | Metrics | Comments（0）

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Efficient Iterative Arbitrary High-Order Methods: an Adaptive Bridge Between Low and High Order Lorenzo Micalizzi, Davide Torlo, Walter Boscheri Communications on Applied Mathematics and Computation    2025, 7 (1): 40-77.   DOI: 10.1007/s42967-023-00290-w Abstract （10）      PDF       Knowledge map    Save We propose a new paradigm for designing efficient p-adaptive arbitrary high-order methods. We consider arbitrary high-order iterative schemes that gain one order of accuracy at each iteration and we modify them to match the accuracy achieved in a specific iteration with the discretization accuracy of the same iteration. Apart from the computational advantage, the newly modified methods allow to naturally perform the p-adaptivity, stopping the iterations when appropriate conditions are met. Moreover, the modification is very easy to be included in an existing implementation of an arbitrary high-order iterative scheme and it does not ruin the possibility of parallelization, if this was achievable by the original method. An application to the Arbitrary DERivative (ADER) method for hyperbolic Partial Differential Equations (PDEs) is presented here. We explain how such a framework can be interpreted as an arbitrary high-order iterative scheme, by recasting it as a Deferred Correction (DeC) method, and how to easily modify it to obtain a more efficient formulation, in which a local a posteriori limiter can be naturally integrated leading to the p-adaptivity and structure-preserving properties. Finally, the novel approach is extensively tested against classical benchmarks for compressible gas dynamics to show the robustness and the computational efficiency. Reference | Related Articles | Metrics | Comments（0）

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Optimal Error Analysis of Linearized Crank-Nicolson Finite Element Scheme for the Time-Dependent Penetrative Convection Problem Min Cao, Yuan Li Communications on Applied Mathematics and Computation    2025, 7 (1): 264-288.   DOI: 10.1007/s42967-023-00269-7 Abstract （15）      PDF       Knowledge map    Save This paper focuses on the optimal error analysis of a linearized Crank-Nicolson finite element scheme for the time-dependent penetrative convection problem, where the mini element and piecewise linear finite element are used to approximate the velocity field, the pressure, and the temperature, respectively. We proved that the proposed finite element scheme is unconditionally stable and the optimal error estimates in L2-norm are derived. Finally, numerical results are presented to confirm the theoretical analysis. Reference | Related Articles | Metrics | Comments（0）

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Discovery of Governing Equations with Recursive Deep Neural Networks Jarrod Mau, Jia Zhao Communications on Applied Mathematics and Computation    2025, 7 (1): 239-263.   DOI: 10.1007/s42967-023-00270-0 Abstract （13）      PDF       Knowledge map    Save Model discovery based on existing data has been one of the major focuses of mathematical modelers for decades. Despite tremendous achievements in model identification from adequate data, how to unravel the models from limited data is less resolved. This paper focuses on the model discovery problem when the data is not efficiently sampled in time, which is common due to limited experimental accessibility and labor/resource constraints. Specifically, we introduce a recursive deep neural network (RDNN) for data-driven model discovery. This recursive approach can retrieve the governing equation efficiently and significantly improve the approximation accuracy by increasing the recursive stages. In particular, our proposed approach shows superior power when the existing data are sampled with a large time lag, from which the traditional approach might not be able to recover the model well. Several examples of dynamical systems are used to benchmark this newly proposed recursive approach. Numerical comparisons confirm the effectiveness of this recursive neural network for model discovery. The accompanying codes are available at https:// github. com/ c2fd/ RDNNs. Reference | Related Articles | Metrics | Comments（0）

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Motion, Dual Quaternion Optimization and Motion Optimization Liqun Qi Communications on Applied Mathematics and Computation    2025, 7 (1): 228-238.   DOI: 10.1007/s42967-023-00262-0 Abstract （12）      PDF       Knowledge map    Save We regard a dual quaternion as a real eight-dimensional vector and present a dual quaternion optimization model. Then we introduce motions as real six-dimensional vectors. A motion means a rotation and a translation. We define a motion operator which maps unit dual quaternions to motions, and a UDQ operator which maps motions to unit dual quaternions. By these operators, we present another formulation of dual quaternion optimization. The objective functions of such dual quaternion optimization models are real valued. They are different from the previous model whose object function is dual number valued. This avoids the two-stage problem, which causes troubles sometimes. We further present an alternative formulation, called motion optimization, which is actually an unconstrained real optimization model. Then we formulate two classical problems in robot research, i.e., the hand-eye calibration problem and the simultaneous localization and mapping (SLAM) problem as such dual quaternion optimization problems as well as such motion optimization problems. This opens a new way to solve these problems. Reference | Related Articles | Metrics | Comments（0）

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On the Order of Accuracy of Edge-Based Schemes: a Peterson-Type Counter-Example Pavel Bakhvalov, Mikhail Surnachev Communications on Applied Mathematics and Computation    2025, 7 (1): 372-391.   DOI: 10.1007/s42967-023-00292-8 Abstract （10）      PDF       Knowledge map    Save Numerical schemes for the transport equation on unstructured meshes usually exhibit the convergence rate p ∈ [k, k + 1], where k is the order of the truncation error. For the discontinuous Galerk in method, the result p = k + 1∕2 is known, and the example where the convergence rate is exactly k + 1∕2 was constructed by Peterson (SIAM J. Numer. Anal. 28: 133–140, 1991) for k = 0 and k = 1. For finite-volume methods with k≥1, there are no theoretical results for general meshes. In this paper, we consider three edge-based finitevolume schemes with k = 1, namely the Barth scheme, the Luo scheme, and the EBR3. For a special family of meshes, under stability assumption we prove the convergence rate p = 3∕2 for the Barth scheme and p = 5∕4 for the other ones. We also present a Petersontype example showing that the values 3∕2 and 5∕4 are optimal. Reference | Related Articles | Metrics | Comments（0）

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The Hermite-Taylor Correction Function Method for Maxwell’s Equations Yann-Meing Law, Daniel Appel? Communications on Applied Mathematics and Computation    2025, 7 (1): 347-371.   DOI: 10.1007/s42967-023-00287-5 Abstract （11）      PDF       Knowledge map    Save The Hermite-Taylor method, introduced in 2005 by Goodrich et al. is highly efficient and accurate when applied to linear hyperbolic systems on periodic domains. Unfortunately, its widespread use has been prevented by the lack of a systematic approach to implementing boundary conditions. In this paper we present the Hermite-Taylor correction function method (CFM), which provides exactly such a systematic approach for handling boundary conditions. Here we focus on Maxwell’s equations but note that the method is easily extended to other hyperbolic problems. Reference | Related Articles | Metrics | Comments（0）

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Numerical Approach for Solving Two-Dimensional Time-Fractional Fisher Equation via HABC-N Method Ren Liu, Lifei Wu Communications on Applied Mathematics and Computation    2025, 7 (1): 315-346.   DOI: 10.1007/s42967-023-00282-w Abstract （11）      PDF       Knowledge map    Save For the two-dimensional time-fractional Fisher equation (2D-TFFE), a hybrid alternating band Crank-Nicolson (HABC-N) method based on the parallel finite difference technique is proposed. The explicit difference method, implicit difference method, and C-N difference method are used simultaneously with the alternating band technique to create the HABC-N method. The existence of the solution and unconditional stability for the HABC-N method, as well as its uniqueness, are demonstrated by theoretical study. The HABC-N method’s convergence order is O(τ2-α + h12 + h22). The theoretical study is bolstered by numerical experiments, which establish that the 2D-TFFE can be solved using the HABC-N method. Reference | Related Articles | Metrics | Comments（0）

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Constructing a CDG Finite Element with Order Two Superconvergence on Rectangular Meshes Xiu Ye, Shangyou Zhang Communications on Applied Mathematics and Computation    2025, 7 (2): 411-425.   DOI: 10.1007/s42967-023-00330-5 Abstract （9）      PDF       Knowledge map    Save A novel conforming discontinuous Galerkin (CDG) finite element method is introduced for Poisson equation on rectangular meshes. This CDG method with discontinuous Pk (k ≥ 1) elements converges to the true solution two orders above the continuous finite element counterpart. Superconvergence of order two for the CDG finite element solution is proved in an energy norm and the L2 norm. A local post-process is defined which lifts a Pk CDG solution to a discontinuous Pk+2 solution. It is proved that the lifted Pk+2 solution converges at the optimal order. The numerical tests illustrate the theoretic findings. Reference | Related Articles | Metrics | Comments（0）

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A Locking-Free and Reduction-Free Conforming Finite Element Method for the Reissner-Mindlin Plate on Rectangular Meshes Shangyou Zhang, Zhimin Zhang Communications on Applied Mathematics and Computation    2025, 7 (2): 470-484.   DOI: 10.1007/s42967-023-00343-0 Abstract （13）      PDF       Knowledge map    Save A family of conforming finite elements are designed on rectangular grids for solving the Reissner-Mindlin plate equation. The rotation is approximated by C1 - Qk+1 in one direction and C0 - Qk in the other direction finite elements. The displacement is approximated by C1 - Qk+1,k+1. The method is locking-free without using any projection/reduction operator. Theoretical proof and numerical confirmation are presented. Reference | Related Articles | Metrics | Comments（0）

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Hierarchical Interpolative Factorization for Self Green’s Function in 3D Modified Poisson-Boltzmann Equations Yihui Tu, Zhenli Xu, Haizhao Yang Communications on Applied Mathematics and Computation    2025, 7 (2): 536-561.   DOI: 10.1007/s42967-023-00352-z Abstract （13）      PDF       Knowledge map    Save The modified Poisson-Boltzmann (MPB) equations are often used to describe the equilibrium particle distribution of ionic systems. In this paper, we propose a fast algorithm to solve the MPB equations with the self Green’s function as the self-energy in three dimensions, where the solution of the self Green’s function poses a computational bottleneck due to the requirement of solving a high-dimensional partial differential equation. Our algorithm combines the selected inversion with hierarchical interpolative factorization for the self Green’s function, building upon our previous result of two dimensions. This approach yields an algorithm with a complexity of O(N log N) by strategically leveraging the locality and low-rank characteristics of the corresponding operators. Additionally, the theoretical O(N) complexity is obtained by applying cubic edge skeletonization at each level for thorough dimensionality reduction. Extensive numerical results are conducted to demonstrate the accuracy and efficiency of the proposed algorithm for problems in three dimensions. Reference | Related Articles | Metrics | Comments（0）

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A Low-Rank Global Krylov Squared Smith Method for Solving Large-Scale Stein Matrix Equation Song Nie, Hua Dai Communications on Applied Mathematics and Computation    2025, 7 (2): 562-588.   DOI: 10.1007/s42967-023-00364-9 Abstract （10）      PDF       Knowledge map    Save This paper deals with the numerical solution of the large-scale Stein and discrete-time Lyapunov matrix equations. Based on the global Arnoldi process and the squared Smith iteration, we propose a low-rank global Krylov squared Smith method for solving largescale Stein and discrete-time Lyapunov matrix equations, and estimate the upper bound of the error and the residual of the approximate solutions for the matrix equations. Moreover, we discuss the restarting of the low-rank global Krylov squared Smith method and provide some numerical experiments to show the efficiency of the proposed method. Reference | Related Articles | Metrics | Comments（0）