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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 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） Select On Error-Based Step Size Control for Discontinuous Galerkin Methods for Compressible Fluid Dynamics Hendrik Ranocha, Andrew R. Winters, Hugo Guillermo Castro, Lisandro Dalcin, Michael Schlottke-Lakemper, Gregor J. Gassner, Matteo Parsani Communications on Applied Mathematics and Computation    2025, 7 (1): 3-39.   DOI: 10.1007/s42967-023-00264-y Abstract （28）      PDF       Knowledge map    Save We study a temporal step size control of explicit Runge-Kutta (RK) methods for compressible computational fluid dynamics (CFD), including the Navier-Stokes equations and hyperbolic systems of conservation laws such as the Euler equations. We demonstrate that error-based approaches are convenient in a wide range of applications and compare them to more classical step size control based on a Courant-Friedrichs-Lewy (CFL) number. Our numerical examples show that the error-based step size control is easy to use, robust, and efficient, e.g., for (initial) transient periods, complex geometries, nonlinear shock capturing approaches, and schemes that use nonlinear entropy projections. We demonstrate these properties for problems ranging from well-understood academic test cases to industrially relevant large-scale computations with two disjoint code bases, the open source Julia packages Trixi.jl with OrdinaryDiffEq.jl and the C/Fortran code SSDC based on PETSc. Reference | Related Articles | Metrics | Comments（0） Select 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） Select 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） Select Characterizations and Properties of Dual Matrix Star Orders Hongxing Wang, Pei Huang Communications on Applied Mathematics and Computation    2025, 7 (1): 179-202.   DOI: 10.1007/s42967-023-00255-z Abstract （15）      PDF       Knowledge map    Save In this paper, we introduce the D-star order, T-star order, and P-star order on the class of dual matrices. By applying the matrix decomposition and dual generalized inverses, we discuss properties, characterizations, and relations among these orders, and illustrate their relations with examples. Reference | Related Articles | Metrics | Comments（0） Select 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） Select 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） Select 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） Select 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 （12）      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） Select 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） Select 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 （12）      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） Select Meshfree Finite Difference Solution of Homogeneous Dirichlet Problems of the Fractional Laplacian Jinye Shen, Bowen Shi, Weizhang Huang Communications on Applied Mathematics and Computation    2025, 7 (2): 589-605.   DOI: 10.1007/s42967-024-00368-z Abstract （12）      PDF       Knowledge map    Save A so-called grid-overlay finite difference method (GoFD) was proposed recently for the numerical solution of homogeneous Dirichlet boundary value problems (BVPs) of the fractional Laplacian on arbitrary bounded domains. It was shown to have advantages of both finite difference (FD) and finite element methods, including their efficient implementation through the fast Fourier transform (FFT) and the ability to work for complex domains and with mesh adaptation. The purpose of this work is to study GoFD in a meshfree setting, a key to which is to construct the data transfer matrix from a given point cloud to a uniform grid. Two approaches are proposed, one based on the moving least squares fitting and the other based on the Delaunay triangulation and piecewise linear interpolation. Numerical results obtained for examples with convex and concave domains and various types of point clouds are presented. They show that both approaches lead to comparable results. Moreover, the resulting meshfree GoFD converges in a similar order as GoFD with unstructured meshes and finite element approximation as the number of points in the cloud increases. Furthermore, numerical results show that the method is robust to random perturbations in the location of the points. Reference | Related Articles | Metrics | Comments（0） Select A Domain Decomposition Method for Nonconforming Finite Element Approximations of Eigenvalue Problems Qigang Liang, Wei Wang, Xuejun Xu Communications on Applied Mathematics and Computation    2025, 7 (2): 606-636.   DOI: 10.1007/s42967-024-00372-3 Abstract （12）      PDF       Knowledge map    Save Since the nonconforming finite elements (NFEs) play a significant role in approximating PDE eigenvalues from below, this paper develops a new and parallel two-level preconditioned Jacobi-Davidson (PJD) method for solving the large scale discrete eigenvalue problems resulting from NFE discretization of 2mth (m = 1, 2) order elliptic eigenvalue problems. Combining a spectral projection on the coarse space and an overlapping domain decomposition (DD), a parallel preconditioned system can be solved in each iteration. A rigorous analysis reveals that the convergence rate of our two-level PJD method is optimal and scalable. Numerical results supporting our theory are given. Reference | Related Articles | Metrics | Comments（0） Select Inverse Lax-Wendroff Boundary Treatment of Discontinuous Galerkin Method for 1D Conservation Laws Lei Yang, Shun Li, Yan Jiang, Chi-Wang Shu, Mengping Zhang Communications on Applied Mathematics and Computation    2025, 7 (2): 796-826.   DOI: 10.1007/s42967-024-00391-0 Abstract （12）      PDF       Knowledge map    Save In this paper, we propose a new class of discontinuous Galerkin (DG) methods for solving 1D conservation laws on unfitted meshes. The standard DG method is used in the interior cells. For the small cut elements around the boundaries, we directly design approximation polynomials based on inverse Lax-Wendroff (ILW) principles for the inflow boundary conditions and introduce the post-processing to preserve the local conservation properties of the DG method. The theoretical analysis shows that our proposed methods have the same stability and numerical accuracy as the standard DG method in the inner region. An additional nonlinear limiter is designed to prevent spurious oscillations if a shock is near the boundary. Numerical results indicate that our methods achieve optimal numerical accuracy for smooth problems and do not introduce additional oscillations in discontinuous problems. Reference | Related Articles | Metrics | Comments（0） Select 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） Select 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） Select Error Estimates of Finite Element Method for the Incompressible Ferrohydrodynamics Equations Shipeng Mao, Jiaao Sun Communications on Applied Mathematics and Computation    2025, 7 (2): 485-535.   DOI: 10.1007/s42967-023-00347-w Abstract （11）      PDF       Knowledge map    Save In this paper, we consider the Shliomis ferrofluid model and study its numerical approximation. We investigate a first-order energy-stable fully discrete finite element scheme for solving the simplified ferrohydrodynamics (SFHD) equations. First, we establish the wellposedness and some regularity results for the solution of the SFHD model. Next we study the Euler semi-implicit time-discrete scheme for the SFHD systems and derive the L2-H1 error estimates for the time-discrete solution. Moreover, certain regularity results for the time-discrete solution are proved rigorously. With the help of these regularity results, we prove the unconditional L2-H1 error estimates for the finite element solution of the SFHD model. Finally, some three-dimensional numerical examples are carried out to demonstrate both the accuracy and efficiency of the fully discrete finite element scheme. Reference | Related Articles | Metrics | Comments（0） Select 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） Select 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） Select 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）