高度选择性的国际数学期刊应用数学计算方法（CMAM）考虑了对计算方法和数值分析的原始数学贡献，其应用主要与偏微分方程相关。CMAM 寻求跨学科的同时保留数值分析的共同点，旨在易于阅读，适用于应用数学领域的广大研究人员。该杂志由 DE GRUYTER 代表白俄罗斯国家科学院数学研究所出版。

Computational Methods in Applied Mathematics is a highly selective international mathematics journal that focuses on computational methods in applied mathematics. The journal welcomes original mathematical contributions related to computational methods and numerical analysis, particularly those closely associated with the application of partial differential equations. It aims to maintain the core elements of numerical analysis while being interdisciplinary, ensuring that the content is easy to read and suitable for a wide range of researchers in the field of applied mathematics. The journal is published by DE GRUYTER on behalf of the Institute of Mathematics of the National Academy of Sciences of Belarus.

《Computational Methods in Applied Mathematics》是一本高度选择性的国际数学期刊，专注于应用数学中的计算方法。该期刊接受与计算方法和数值分析相关的原创数学贡献，特别是与偏微分方程的应用密切相关。期刊旨在跨学科的同时，保持数值分析的核心要素，力求内容易于阅读，适合广大应用数学领域的研究人员。该杂志由DE GRUYTER出版，代表白俄罗斯国家科学院数学研究所。

Computational Methods in Applied Mathematics

Abstract We present in this paper a rigorous error analysis of the vector penalty-projection method for solving the time-dependent incompressible Stokes equations with open boundary conditions on part of the boundary. First, we prove the stability of the scheme. Then we provide an error analysis for the second-order vector penalty-projection method which shows that the convergence rate of the error on the velocity and the pressure is of order 2 in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>l</m:mi> <m:mi mathvariant="normal">∞</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mi mathvariant="bold">L</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> l^{\infty}(\mathbf{L}^{2}(\Omega)) and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>l</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> l^{2}(L^{2}(\Omega)) respectively. In addition, it is shown that the splitting errors of the method varies as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">O</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ε</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathcal{O}(\varepsilon) , where 𝜀 is a penalty parameter chosen as small as desired. Several numerical tests in agreement with the theoretical results are presented. To the best of our knowledge, this paper provides the first rigorous proof of optimal error estimates for second-order splitting schemes with open boundary conditions.

Rima Cheaytou

Philippe Angot

Error Analysis of the Vector Penalty-Projection Methods for the Time-Dependent Stokes Equations with Open Boundary Conditions

带有开边界条件的时变斯托克斯方程的向量罚投影方法误差分析

Abstract This paper aims to give a unified construction framework of meshless structure-preserving algorithms to solve the d -dimensional ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>d</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {d=1} or 2) nonlinear Schrödinger equation. Based on the method of lines, we first derive a finite-dimensional Hamiltonian system by using the radial basis function method of the quasi-interpolation and the technique of left-multiplying a diagonal matrix to discretize the space direction. Then suitable geometric numerical integrations can be used to discretize the time direction, which yields a class of meshless structure-preserving algorithms. In addition to the construction, the structure-preserving properties and their proofs are also provided in detail. Besides the uniform and nonuniform grids, the numerical experiments on the random grids are also emphasized to verify the theoretical research well, which is of great significance for scattering points based on the characteristics of actual problems.

Jialing Wang

Zhengting Zhou

Zhoujin Lin

A Class of Meshless Structure-Preserving Algorithms for the Nonlinear Schrödinger Equation

一类无网格保结构算法用于非线性薛定谔方程

Abstract In this study, we develop and implement numerical schemes to solve classes of two-dimensional fourth-order partial differential equations. These methods are fourth-order accurate in space and second-order accurate in time and require only nine spatial grid points of a single compact cell. The proposed discretizations allow the use of Dirichlet boundary conditions only without the need to discretize the derivative boundary conditions and thus avoids the use of ghost points. No transformation or linearization technique is used to handle nonlinearity and the obtained block tri-diagonal nonlinear system has been solved by Newton’s block iteration method. It is discussed how our formulation is able to tackle linear singular problems and it is ensured that the methods retain their orders and accuracy everywhere in the solution region. The proposed two-level method is shown to be unconditionally stable for a class of two-dimensional fourth-order linear parabolic equation. We also discuss the alternating direction implicit (ADI) method for solving two-dimensional fourth-order linear parabolic equation. The proposed difference methods has been successfully tested on the two-dimensional vibration problem, Boussinesq equation, extended Fisher–Kolmogorov equation and Kuramoto–Sivashinsky equation. Numerical results demonstrate that the schemes are highly accurate in solving a large class of physical problems.

Ranjan Kumar Mohanty

Deepti Kaur

High Accuracy Compact Operator Methods for Two-Dimensional Fourth Order Nonlinear Parabolic Partial Differential Equations

二维四阶非线性抛物型偏微分方程的高精度紧致算子方法

Abstract This article develops and analyses a conforming virtual element scheme for the spatial discretization of parabolic integro-differential equations combined with backward Euler’s scheme for temporal discretization. With the help of Ritz–Voltera and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> L^{2} projection operators, optimal a priori error estimates are established. Moreover, several numerical experiments are presented to confirm the computational efficiency of the proposed scheme and validate the theoretical findings.

Sangita Yadav

Meghana Suthar

Sarvesh Kumar

A Conforming Virtual Element Method for Parabolic Integro-Differential Equations

用于抛物型积分-微分方程的协调虚拟元方法

Abstract We present an adaptive absorbing boundary layer technique for the nonlinear Schrödinger equation that is used in combination with the Time-splitting Fourier spectral method (TSSP) as the discretization for the NLS equations. We propose a new complex absorbing potential (CAP) function based on high order polynomials, with the major improvement that an explicit formula for the coefficients in the potential function is employed for adaptive parameter selection. This formula is obtained by an extension of the analysis in [R. Kosloff and D. Kosloff, Absorbing boundaries for wave propagation problems, J. Comput. Phys. 63 1986, 2, 363–376]. We also show that our imaginary potential function is more efficient than what is used in the literature. Numerical examples show that our ansatz is significantly better than existing approaches. We show that our approach can very accurately compute the solutions of the NLS equations in one dimension, including in the case of multi-dominant wave number solutions.

Hans Peter Stimming

Xin Wen

Norbert J. Mauser

Adaptive Absorbing Boundary Layer for the Nonlinear Schrödinger Equation

非线性薛定谔方程的自适应吸收边界层

Abstract In [L. Bourhrara, A new numerical method for solving the Boltzmann transport equation using the PN method and the discontinuous finite elements on unstructured and curved meshes, J. Comput. Phys. 397 2019, Article ID 108801], a numerical scheme based on a combined spherical harmonics and discontinuous Galerkin finite element method for the resolution of the Boltzmann transport equation is proposed. One of its features is that a streamline weight is added to the test function to obtain the variational formulation. In the present paper, restricting our attention to the advective part of the Boltzmann equation, we prove the convergence and provide error estimates of this numerical scheme. To this end, the original variational formulation is restated in a broken functional space. The use of broken functional spaces enables to build a conforming approximation, that is the finite element space is a subspace of the broken functional space. The setting of a conforming approximation simplifies the numerical analysis, in particular the error estimates, for which a Céa’s type lemma and standard interpolation estimates are sufficient for our analysis. For our numerical scheme, based on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℙ</m:mi> <m:mi>k</m:mi> </m:msup> </m:math> {\mathbb{P}^{k}} discontinuous Galerkin finite elements (in space) on a mesh of size h and a spherical harmonics approximation of order N (in the angular variable), the convergence rate is of order <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">𝒪</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mi>N</m:mi> <m:mrow> <m:mo>-</m:mo> <m:mi>t</m:mi> </m:mrow> </m:msup> <m:mo>+</m:mo> <m:msup> <m:mi>h</m:mi> <m:mi>k</m:mi> </m:msup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathcal{O}(N^{-t}+h^{k})} for a smooth solution which admits partial derivatives of order <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>k</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {k+1} and t with respect to the spatial and angular variables, respectively. For <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>k</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {k=0} (piecewise constant finite elements) we also obtain a convergence result of order <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">𝒪</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mi>N</m:mi> <m:mrow> <m:mo>-</m:mo> <m:mi>t</m:mi> </m:mrow> </m:msup> <m:mo>+</m:mo> <m:msup> <m:mi>h</m:mi> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> </m:msup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathcal{O}(N^{-t}+h^{\frac{1}{2}})} . Numerical experiments in one, two and three dimensions are provided, showing a better convergence behavior for the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> {L^{2}} -norm, typically of one more order, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">𝒪</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mi>N</m:mi> <m:mrow> <m:mo>-</m:mo> <m:mi>t</m:mi> </m:mrow> </m:msup> <m:mo>+</m:mo> <m:msup> <m:mi>h</m:mi> <m:mrow> <m:mi>k</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathcal{O}(N^{-t}+h^{k+1})} .

Kenneth Assogba

Grégoire Allaire

Lahbib Bourhrara

Analysis of a Combined Spherical Harmonics and Discontinuous Galerkin Discretization for the Boltzmann Transport Equation

结合球谐函数与间断伽辽金离散化的玻尔兹曼输运方程分析

Abstract We consider nodal-based Lagrangian interpolations for the finite element approximation of the Maxwell eigenvalue problem. The first approach introduced is a standard Galerkin method on Powell–Sabin meshes, which has recently been shown to yield convergent approximations in two dimensions, whereas the other two are stabilized formulations that can be motivated by a variational multiscale approach. For the latter, a mixed formulation equivalent to the original problem is used, in which the operator has a saddle point structure. The Lagrange multiplier introduced to enforce the divergence constraint vanishes in an appropriate functional setting. The first stabilized method consists of an augmented formulation including a mesh dependent term that can be regarded as the Laplacian of the divergence constraint multiplier. The second formulation is based on orthogonal projections, which can be recast as a residual based stabilization technique. We rely on the classical spectral theory to analyze the approximating methods for the eigenproblem. The stability and convergence aspects are inherited from the associated source problems together with an assumption which is discussed numerically. We investigate the performance of the proposed formulations and provide some convergence results validating the theoretical ones for several benchmark tests, including ones with smooth and singular solutions.

Daniele Boffi

Ramon Codina

Önder Türk

Finite Element Formulations for Maxwell’s Eigenvalue Problem Using Continuous Lagrangian Interpolations

用于麦克斯韦特征值问题的有限元公式化方法：采用连续拉格朗日插值

Abstract The P2 BDM H-div finite element is enriched by five <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>P</m:mi> <m:mn>5</m:mn> </m:msub> </m:math> P_{5} divergence-free bubbles on each triangle. The resulting finite element remains H-div and is also H-curl nonconforming. Thus the new finite element combined with the discontinuous <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>P</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> P_{1} element for pressure, is stabilizer-free and still divergence-free in solving the Stokes equations on triangular meshes. The stabilization can also be done by three, four, or six <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>P</m:mi> <m:mn>5</m:mn> </m:msub> </m:math> P_{5} divergence-free bubble-enrichment each triangle, shown both in theory and in computation. Numerical tests show the advantage of the divergence-free element over the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>P</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> P_{2} Taylor–Hood finite element.

Abbreviation	J. Comput. Methods. Appl. Math.
Journal Impact	1.16
Quartiles(Global)	MATHEMATICS, APPLIED(Q3)

ISSN	1609-4840, 1609-9389
h-index	33

Publication Information	Publisher: Walter de Gruyter GmbH，Publishing cycle: ，Journal Type: journal，Open Access Journals: No
Basic data	Year of publication: 2001，Proportion of original research papers: 100.00%，Self Citation Rate:10.00%， Gold OA Rate: 11.24%

Computational Methods in Applied Mathematics

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Error Analysis of the Vector Penalty-Projection Methods for the Time-Dependent Stokes Equations with Open Boundary Conditions

A Class of Meshless Structure-Preserving Algorithms for the Nonlinear Schrödinger Equation

High Accuracy Compact Operator Methods for Two-Dimensional Fourth Order Nonlinear Parabolic Partial Differential Equations

A Conforming Virtual Element Method for Parabolic Integro-Differential Equations

Adaptive Absorbing Boundary Layer for the Nonlinear Schrödinger Equation

A Class of Meshless Structure-Preserving Algorithms for the Nonlinear Schrödinger Equation

Error Analysis of the Vector Penalty-Projection Methods for the Time-Dependent Stokes Equations with Open Boundary Conditions

Analysis of a Combined Spherical Harmonics and Discontinuous Galerkin Discretization for the Boltzmann Transport Equation

Finite Element Formulations for Maxwell’s Eigenvalue Problem Using Continuous Lagrangian Interpolations

A P ₂ H-Div-Nonconforming-H-Curl Finite Element for the Stokes Equations on Triangular Meshes

Computational Methods in Applied Mathematics

Journal Citation Format

A journal article with 1 author

A journal article with 2 authors

A journal article with 3 authors

A journal article with 5 or more authors

Books Citation Format

Thesis Citation Format

Web sites Citation Format

Patent Citation Format

Staying up late manually editing references? ivySCI automatically matches journals and helps you generate references with a single click.

Share Submission Experience

Error Analysis of the Vector Penalty-Projection Methods for the Time-Dependent Stokes Equations with Open Boundary Conditions

A Class of Meshless Structure-Preserving Algorithms for the Nonlinear Schrödinger Equation

High Accuracy Compact Operator Methods for Two-Dimensional Fourth Order Nonlinear Parabolic Partial Differential Equations

A Conforming Virtual Element Method for Parabolic Integro-Differential Equations

Adaptive Absorbing Boundary Layer for the Nonlinear Schrödinger Equation

A Class of Meshless Structure-Preserving Algorithms for the Nonlinear Schrödinger Equation

Error Analysis of the Vector Penalty-Projection Methods for the Time-Dependent Stokes Equations with Open Boundary Conditions

Analysis of a Combined Spherical Harmonics and Discontinuous Galerkin Discretization for the Boltzmann Transport Equation

Finite Element Formulations for Maxwell’s Eigenvalue Problem Using Continuous Lagrangian Interpolations

A P 2 H-Div-Nonconforming-H-Curl Finite Element for the Stokes Equations on Triangular Meshes

A P ₂ H-Div-Nonconforming-H-Curl Finite Element for the Stokes Equations on Triangular Meshes