Applied Numerical Mathematics最新文献

{"title":"An energy stable and well-balanced scheme for the Ripa system","authors":"K.R. Arun ,&nbsp;R. Ghorai","doi":"10.1016/j.apnum.2025.05.008","DOIUrl":"10.1016/j.apnum.2025.05.008","url":null,"abstract":"<div><div>We design and analyse an energy-stable, structure-preserving, and well-balanced scheme for the Ripa system of shallow water equations. The energy stability of the numerical solutions is achieved by introducing appropriate stabilisation terms in the discretisation of the convective fluxes of mass and momentum, the pressure gradient, and the topography source term. The careful selection of the interface values for the water height and temperature ensures the scheme's well-balancing property for three physically relevant hydrostatic steady states. The scheme, which is explicit in time and finite volume in space, preserves the positivity of both the water height and the temperature, and it is weakly consistent with the continuous model equations in the sense of Lax-Wendroff. Additionally, a suitable modification of the source term discretisation and timestep criterion allows the scheme to handle wet/dry fronts in equilibrium. The results of extensive numerical case studies on benchmark test problems confirm the theoretical findings.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"216 ","pages":"Pages 187-209"},"PeriodicalIF":2.2,"publicationDate":"2025-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144134118","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"High-order and mass-conservative regularized implicit-explicit relaxation Runge-Kutta methods for the low regularity Schrödinger equations","authors":"Jingye Yan ,&nbsp;Hong Zhang ,&nbsp;Yabing Wei ,&nbsp;Xu Qian","doi":"10.1016/j.apnum.2025.05.009","DOIUrl":"10.1016/j.apnum.2025.05.009","url":null,"abstract":"<div><div>The non-differentiability of the singular nonlinearities (<span><math><mi>f</mi><mo>=</mo><mi>ln</mi><mo>⁡</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span> and <span><math><mi>f</mi><mo>=</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn><mi>α</mi></mrow></msup><mo>,</mo><mspace></mspace><mi>α</mi><mo>&lt;</mo><mn>0</mn></math></span>) at <span><math><mi>u</mi><mo>=</mo><mn>0</mn></math></span> brings significant challenges in designing accurate and efficient numerical schemes for the low regularity Schrödinger equations (LorSE). In order to address the singularity, we propose an energy regularization for the LorSE. For the regularized models, we apply Implicit-explicit Relaxation Runge-Kutta methods which are linearly implicit, high order and mass-conserving for temporal discretization, in conjunction with the Fourier pseudo-spectral method in space. Ultimately, numerical results are presented to validate the efficiency of the proposed methods.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"216 ","pages":"Pages 210-221"},"PeriodicalIF":2.2,"publicationDate":"2025-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144139588","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Numerical analysis for an evolution equation with the p-biharmonic operator","authors":"Rui M.P. Almeida ,&nbsp;José C.M. Duque ,&nbsp;Jorge Ferreira ,&nbsp;Willian S. Panni","doi":"10.1016/j.apnum.2025.05.006","DOIUrl":"10.1016/j.apnum.2025.05.006","url":null,"abstract":"<div><div>In this paper, we consider a parabolic equation with the <em>p</em>-biharmonic operator, where <span><math><mi>p</mi><mo>&gt;</mo><mn>1</mn></math></span>. By employing a suitable change of variable, we transform the fourth-order nonlinear parabolic problem into a system of two second-order differential equations. We investigate the properties of the discretized solution in spatial and temporal variables. Using the Brouwer fixed point theorem we prove the existence of the discretized solution. Through classical functional analysis techniques we demonstrate the uniqueness and a priori estimates of the discretized solution. We establish the order of convergence in space and time, we establish the relationship between the temporal variable and the spatial variable, ensuring the existence of the convergence order. Additionally, we highlight that the change in variable carried out is extremely advantageous, as it allows us to obtain the order of convergence for the solution and its higher order derivatives using only lower-degree polynomials. Finally, using the finite element method with Lagrange basis, we implement the computational codes in Matlab software, considering the one and two-dimensional cases. We present three examples to illustrate and validate the theory.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"216 ","pages":"Pages 164-186"},"PeriodicalIF":2.2,"publicationDate":"2025-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144134117","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Sinc approximation method for solving system of singularly perturbed parabolic convection-diffusion equations","authors":"N. Barzehkar,&nbsp;A. Barati,&nbsp;R. Jalilian","doi":"10.1016/j.apnum.2025.05.005","DOIUrl":"10.1016/j.apnum.2025.05.005","url":null,"abstract":"<div><div>In this paper, the Sinc-collocation method is used to solve singularly perturbed parabolic convection-diffusion system. The convergence analysis of the proposed method is discussed, it is shown that the convergence of the method is at an exponential rate in space dimension. Finally, some numerical results are given to validate the theoretical results. Also, the obtained results show the accuracy and efficiency of the method compared with other methods.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"216 ","pages":"Pages 127-139"},"PeriodicalIF":2.2,"publicationDate":"2025-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144116814","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A study of Cauchy problem of the Helmholtz equation based on a relaxation model: Regularization and analysis","authors":"Rongfang Gong ,&nbsp;Xiaohui Liu ,&nbsp;Catharine W.K. Lo ,&nbsp;Gaocheng Yue","doi":"10.1016/j.apnum.2025.05.007","DOIUrl":"10.1016/j.apnum.2025.05.007","url":null,"abstract":"<div><div>In this paper, we consider a Cauchy problem of the Helmholtz equation of recovering both missing voltage and current on inaccessible boundary from Cauchy data measured on the remaining accessible boundary. With an introduction of a relaxation parameter, the Dirichlet boundary conditions are approximated by two Robin ones. Associated with two mixed boundary value problems, a regularized Kohn-Vogelius formulation is proposed. Compared to the existing work, weaker regularity is required on the Dirichlet data and no Dirichlet BVPs needs to be solved. This makes the proposed model simpler and more efficient in computation. The well-posedness analysis about the relaxation model and error estimates of the corresponding inverse problem are obtained. A series of theoretical results are established for the new reconstruction model. Several numerical examples are provided to show feasibility and effectiveness of the proposed method.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"216 ","pages":"Pages 140-163"},"PeriodicalIF":2.2,"publicationDate":"2025-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144116617","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A second-order semi-implicit spectral deferred correction scheme for Cahn-Hilliard-Navier-Stokes equation","authors":"Xin Liu ,&nbsp;Dandan Xue ,&nbsp;Shuaichao Pei ,&nbsp;Hong Yang","doi":"10.1016/j.apnum.2025.05.003","DOIUrl":"10.1016/j.apnum.2025.05.003","url":null,"abstract":"<div><div>In this paper, a second-order and energy stable numerical scheme is developed to solve the Cahn-Hilliard-Navier-Stokes phase field model with matching density. This scheme is based on the second-order semi-implicit spectral deferred correction method and the energy stable first-order convex splitting approach. A fully discretized scheme with finite elements for the spatial discretization is developed to solve this coupled system. The energy stability of our scheme is theoretically proven, and its convergence is verified numerically. Numerical experiments are conducted to demonstrate the stability and reliability of the proposed scheme.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"216 ","pages":"Pages 39-55"},"PeriodicalIF":2.2,"publicationDate":"2025-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143946841","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On greedy randomized coordinate updating iteration methods for solving symmetric eigenvalue problems","authors":"Zhong-Zhi Bai","doi":"10.1016/j.apnum.2025.04.010","DOIUrl":"10.1016/j.apnum.2025.04.010","url":null,"abstract":"<div><div>In order to compute the smallest eigenvalue and its corresponding eigenvector of a large-scale, real, and symmetric matrix, we propose a class of greedy randomized coordinate updating iteration methods based on the principle that the indices of larger entries in absolute value of the current residual are selected with a higher probability and, with respect to this index set, the next iterate is updated from the current iterate along with the selected coordinate such that the corresponding entry of the residual is annihilated, resulting in fast convergence rates of the proposed iteration methods. Under appropriate conditions, we prove the convergence of both sequences of the Rayleigh-quotients and the acute angles between the iterates and the eigenvector in terms of the expectation. By numerical experiments, we show that this class of greedy randomized coordinate updating iteration methods are advantageous over the parameterized power method and the coordinate descent method in both iteration counts and computing times. Moreover, with theoretical analysis and computational performance, we confirm that the convergence property of this class of iteration methods can be improved significantly by suitably choosing the arbitrary nonnegative parameter involved in the greedy probability criterion.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"216 ","pages":"Pages 76-97"},"PeriodicalIF":2.2,"publicationDate":"2025-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144069908","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Data-dependent moving least squares","authors":"David Levin ,&nbsp;José M. Ramón ,&nbsp;Juan Ruiz-Álvarez ,&nbsp;Dionisio F. Yáñez","doi":"10.1016/j.apnum.2025.05.002","DOIUrl":"10.1016/j.apnum.2025.05.002","url":null,"abstract":"<div><div>In this paper, we address a data-dependent modification of the moving least squares (MLS) problem. We propose a novel approach by replacing the traditional weight functions with new functions that assign smaller weights to nodes that are close to discontinuities, while still assigning smaller weights to nodes that are far from the point of approximation. Through this adjustment, we are able to mitigate the undesirable Gibbs phenomenon that appears close to the discontinuities in the classical MLS approach, and reduce the smearing of discontinuities in the final approximation of the original data. The core of our method involves accurately identifying those nodes affected by the presence of discontinuities using smoothness indicators, a concept derived from the data-dependent WENO method. Our formulation results in a data-dependent weighted least squares problem where the weights depend on two factors: the distances between nodes and the point of approximation, and the smoothness of the data in a region of predetermined radius around the nodes. We explore the design of the new data-dependent approximant, analyze its properties including polynomial reproduction, accuracy, and smoothness, and study its impact on diffusion and the Gibbs phenomenon. Numerical experiments are conducted to validate the theoretical findings, and we conclude with some insights and potential directions for future research.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"216 ","pages":"Pages 56-75"},"PeriodicalIF":2.2,"publicationDate":"2025-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143946842","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A posteriori error estimates and time adaptivity for fully discrete finite element method for the incompressible Navier-Stokes equations","authors":"Shuo Yang,&nbsp;Hongjiong Tian","doi":"10.1016/j.apnum.2025.05.001","DOIUrl":"10.1016/j.apnum.2025.05.001","url":null,"abstract":"<div><div>In this paper, we study a posteriori error estimates for the incompressible Navier-Stokes equations in a convex polygonal domain. The semi-implicit variable step-size two-step backward differentiation formula (BDF2) is employed for the time discretization and the Taylor–Hood finite element method (FEM) is used for the space discretization. We prove energy stability of semi-implicit variable step-size BDF2 FEM under different Courant Friedreich Lewy (CFL)-type conditions by utilizing different embeddings for the nonlinear term. Two appropriate reconstructions of the approximate solution are proposed to obtain the time discretization error. Resorting to the energy stability and the quadratic reconstructions, we obtain a posteriori upper and lower error bounds for the fully discrete approximation. We further develop a time adaptive algorithm for efficient time step control based on the time error estimators. Several numerical experiments are performed to verify our theoretical results and demonstrate the efficiency of the time adaptive algorithm.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"216 ","pages":"Pages 17-38"},"PeriodicalIF":2.2,"publicationDate":"2025-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143936933","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Numerical analysis of American option pricing in a two-asset jump-diffusion model","authors":"Hao Zhou,&nbsp;Duy-Minh Dang","doi":"10.1016/j.apnum.2025.03.005","DOIUrl":"10.1016/j.apnum.2025.03.005","url":null,"abstract":"<div><div>This paper addresses an important gap in rigorous numerical treatments for pricing American options under correlated two-asset jump-diffusion models using the viscosity solution framework, with a particular focus on the Merton model. The pricing of these options is governed by complex two-dimensional (2-D) variational inequalities that incorporate cross-derivative terms and nonlocal integro-differential terms due to the presence of jumps. Existing numerical methods, primarily based on finite differences, often struggle with preserving monotonicity in the approximation of cross-derivatives–a key requirement for ensuring convergence to the viscosity solution. In addition, these methods face challenges in accurately discretizing 2-D jump integrals.</div><div>We introduce a novel approach to effectively tackle the aforementioned variational inequalities while seamlessly handling cross-derivative terms and nonlocal integro-differential terms through an efficient and straightforward-to-implement monotone integration scheme. Within each timestep, our approach explicitly enforces the inequality constraint, resulting in a 2-D Partial Integro-Differential Equation (PIDE) to solve. Its solution is then expressed as a 2-D convolution integral involving the Green's function of the PIDE. We derive an infinite series representation of this Green's function, where each term is strictly positive and computable. This series facilitates the numerical approximation of the PIDE solution through a monotone integration method, such as the composite quadrature rule. To further enhance efficiency, we develop an efficient implementation of this monotone integration scheme via Fast Fourier Transforms, exploiting the Toeplitz matrix structure.</div><div>The proposed method is proved to be both <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span>-stable and consistent in the viscosity sense, ensuring its convergence to the viscosity solution of the variational inequality. Extensive numerical results validate the effectiveness and robustness of our approach, highlighting its practical applicability and theoretical soundness.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"216 ","pages":"Pages 98-126"},"PeriodicalIF":2.2,"publicationDate":"2025-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144069909","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}