Advances in Computational Mathematics最新文献

Noniterative localized exponential time differencing methods for hyperbolic conservation laws 双曲型守恒律的非迭代局域指数差分方法

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2025-05-27 DOI: 10.1007/s10444-025-10240-0

Cao-Kha Doan, Phuoc-Toan Huynh, Thi-Thao-Phuong Hoang

{"title":"Noniterative localized exponential time differencing methods for hyperbolic conservation laws","authors":"Cao-Kha Doan, Phuoc-Toan Huynh, Thi-Thao-Phuong Hoang","doi":"10.1007/s10444-025-10240-0","DOIUrl":"10.1007/s10444-025-10240-0","url":null,"abstract":"<div><p>The paper is concerned with efficient time discretization methods based on exponential integrators for scalar hyperbolic conservation laws. The model problem is first discretized in space by the discontinuous Galerkin method, resulting in a system of nonlinear ordinary differential equations. To solve such a system, exponential time differencing of order 2 (ETDRK2) is employed with Jacobian linearization at each time step. The scheme is fully explicit and relies on the computation of matrix exponential vector products. To accelerate such computation, we further construct a noniterative, nonoverlapping domain decomposition algorithm, namely localized ETDRK2, which loosely decouples the system at each time step via suitable interface conditions. Temporal error analysis of the proposed global and localized ETDRK2 schemes is rigorously proved; moreover, the schemes are shown to be conservative under periodic boundary conditions. Numerical results for the Burgers’ equation in one and two dimensions (with moving shocks) are presented to verify the theoretical results and illustrate the performance of the global and localized ETDRK2 methods where large time step sizes can be used without affecting numerical stability.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 3","pages":""},"PeriodicalIF":1.7,"publicationDate":"2025-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144140275","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Linearly implicit and large time-stepping conservative exponential relaxation schemes for the nonlocal cubic Gross-Pitaevskii equation 非局部三次Gross-Pitaevskii方程的线性隐式和大时步保守指数松弛格式

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2025-05-27 DOI: 10.1007/s10444-025-10238-8

Yayun Fu, Xu Qian, Songhe Song, Dongdong Hu

{"title":"Linearly implicit and large time-stepping conservative exponential relaxation schemes for the nonlocal cubic Gross-Pitaevskii equation","authors":"Yayun Fu, Xu Qian, Songhe Song, Dongdong Hu","doi":"10.1007/s10444-025-10238-8","DOIUrl":"10.1007/s10444-025-10238-8","url":null,"abstract":"<div><p>The nonlocal cubic Gross-Pitaevskii equation, in comparison to the cubic Gross-Pitaevskii equation, incorporates a nonlocal diffusion operator and can capture a wider range of practical phenomena. However, this nonlocal formulation significantly increases the computational expenses in numerical simulations, necessitating the development of efficient and accurate time integration schemes. This paper uses the relaxation method to present two linearly implicit conservative exponential schemes for the nonlocal cubic Gross-Pitaevskii equation. One proposed scheme can inherit the discrete energy while the other preserves the mass in the discrete scene. We first apply the Fourier pseudo-spectral method to the equation and derive a conservative semi-discrete system. Then, based on the ideas of the traditional relaxation method, adopting the exponential time difference method to approximate the system in time can lead to an energy-preserving exponential scheme. The mass-preserving scheme is derived by using the integral factor method to discretize the system in the temporal direction. The stability results of the constructed schemes are given. In addition, all schemes are linearly implicit and can be implemented efficiently with a large time step. Finally, numerical results show that both proposed methods are remarkably efficient and have better stability than the original relaxation scheme.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 3","pages":""},"PeriodicalIF":1.7,"publicationDate":"2025-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144140276","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A hybrid boundary integral-PDE approach for the approximation of the demagnetization potential in micromagnetics 微磁学中退磁势的边界积分-偏微分方程混合逼近方法

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2025-05-15 DOI: 10.1007/s10444-025-10233-z

Doghonay Arjmand, Víctor Martínez Calzada

{"title":"A hybrid boundary integral-PDE approach for the approximation of the demagnetization potential in micromagnetics","authors":"Doghonay Arjmand, Víctor Martínez Calzada","doi":"10.1007/s10444-025-10233-z","DOIUrl":"10.1007/s10444-025-10233-z","url":null,"abstract":"<div><p>The demagnetization field in micromagnetism is given as the gradient of a potential that solves a partial differential equation (PDE) posed in <span>(mathbb {R}^d)</span>. In its most general form, this PDE is supplied with continuity condition on the boundary of the magnetic domain, and the equation includes a discontinuity in the gradient of the potential over the boundary. Typical numerical algorithms to solve this problem rely on the representation of the potential via the Green’s function, where a volume and a boundary integral terms need to be accurately approximated. From a computational point of view, the volume integral dominates the computational cost and can be difficult to approximate due to the singularities of the Green’s function. In this article, we propose a hybrid model, where the overall potential can be approximated by solving two uncoupled PDEs posed in bounded domains, whereby the boundary conditions of one of the PDEs are obtained by a low cost boundary integral. Moreover, we provide a convergence analysis of the method under two separate theoretical settings: periodic magnetization and high-frequency magnetization. Numerical examples are given to verify the convergence rates.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 3","pages":""},"PeriodicalIF":1.7,"publicationDate":"2025-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143949593","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

WKB-based third order method for the highly oscillatory 1D stationary Schrödinger equation 基于wkb的三阶方法求解高振荡一维平稳Schrödinger方程

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2025-05-15 DOI: 10.1007/s10444-025-10234-y

Anton Arnold, Jannis Körner

引用次数: 0

Error analysis of a hybrid numerical method for optimal control problem governed by parabolic PDEs in random cylindrical domains 随机圆柱域抛物型偏微分方程最优控制问题的混合数值方法误差分析

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2025-05-13 DOI: 10.1007/s10444-025-10237-9

Mengya Feng, Tongjun Sun

引用次数: 0

Stable approximate evaluation of unbounded matrix operator and its application to an inverse problem 无界矩阵算子的稳定近似求值及其在逆问题中的应用

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2025-05-09 DOI: 10.1007/s10444-025-10235-x

Shuang Yu, Hongqi Yang

引用次数: 0

Efficient algorithms for Tucker decomposition via approximate matrix multiplication 基于近似矩阵乘法的高效塔克分解算法

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2025-04-22 DOI: 10.1007/s10444-025-10232-0

Maolin Che, Yimin Wei, Hong Yan

引用次数: 0

Computing the action of the matrix generating function of Bernoulli polynomials on a vector with an application to non-local boundary value problems 计算伯努利多项式的矩阵生成函数对向量的作用，并应用于非局部边值问题

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2025-04-10 DOI: 10.1007/s10444-025-10231-1

Lidia Aceto, Luca Gemignani

引用次数: 0

A discontinuous plane wave neural network method for Helmholtz equation and time-harmonic Maxwell’s equations 求解Helmholtz方程和时谐Maxwell方程的不连续平面波神经网络方法

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2025-04-07 DOI: 10.1007/s10444-025-10229-9

Long Yuan, Qiya Hu

{"title":"A discontinuous plane wave neural network method for Helmholtz equation and time-harmonic Maxwell’s equations","authors":"Long Yuan, Qiya Hu","doi":"10.1007/s10444-025-10229-9","DOIUrl":"10.1007/s10444-025-10229-9","url":null,"abstract":"<div><p>In this paper, we propose a <i>discontinuous</i> plane wave neural network (DPWNN) method with <span>(hp-)</span>refinement for approximately solving Helmholtz equation and time-harmonic Maxwell equations. In this method, we define a quadratic functional as in the plane wave least square (PWLS) method with <span>(h-)</span>refinement and introduce new discretization sets spanned by element-wise neural network functions with a single hidden layer, where the activation function on each element is chosen as a complex-valued exponential function like the plane wave function. The desired approximate solution is recursively generated by iteratively solving a quasi-minimization problem associated with the functional and the sets described above, which is defined by a sequence of approximate minimizers of the underlying residual functionals, where plane wave direction angles and activation coefficients are alternatively computed by iterative algorithms. For the proposed DPWNN method, the plane wave directions are adaptively determined in the iterative process, which is different from that in the standard PWLS method (where the plane wave directions are preliminarily given). Numerical experiments will confirm that this DPWNN method can generate approximate solutions with higher accuracy than the PWLS method.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 2","pages":""},"PeriodicalIF":1.7,"publicationDate":"2025-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143786662","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Low-rank exponential integrators for stiff differential Riccati equations 刚性Riccati微分方程的低秩指数积分器

IF 1.7 3区数学

Advances in Computational Mathematics Pub Date : 2025-04-02 DOI: 10.1007/s10444-025-10228-w

Hao Chen, Alfio Borzì

引用次数: 0