Computers & Mathematics with Applications最新文献

{"title":"Energy-stable linear convex splitting methods for the parabolic sine-Gordon equation","authors":"Minhwan Ji,&nbsp;Jaemin Shin","doi":"10.1016/j.camwa.2025.08.007","DOIUrl":"10.1016/j.camwa.2025.08.007","url":null,"abstract":"<div><div>We propose a linear convex splitting approach for the parabolic sine-Gordon equation. This linear formulation ensures unique solvability and high computational efficiency. When combined with a convex splitting Runge–Kutta method, it achieves high-order temporal accuracy and unconditional energy stability. For the first-order scheme, we establish the discrete maximum principle, a notable property of the parabolic sine-Gordon equation, although this principle is observed to be numerically violated in the second-order scheme. Spatial discretization is performed employing a standard second-order accurate finite difference method. Numerical experiments are provided to validate the accuracy, energy stability, and dynamic behavior of the proposed schemes.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"198 ","pages":"Pages 24-37"},"PeriodicalIF":2.5,"publicationDate":"2025-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144813907","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":"Immersed finite element methods for linear and quasi-linear elliptic interface problems","authors":"Slimane Adjerid","doi":"10.1016/j.camwa.2025.08.001","DOIUrl":"10.1016/j.camwa.2025.08.001","url":null,"abstract":"<div><div>We present a framework for constructing immersed finite element (IFE) spaces for second-order elliptic interface problems on interface-independent meshes exhibiting optimal convergence rates of <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>h</mi></mrow><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> in the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> norm and <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>h</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>)</mo></math></span> in the <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> norm using a polynomial degree <em>p</em>. We consider linear problems with variable coefficients as well as quasi-linear problems. We present algorithms to create IFE spaces on elements cut by the interface and algorithms to solve the resulting finite element problems. We also present numerical results to demonstrate the performance of the IFE method for several linear problems with variable coefficients and quasi-linear elliptic interface problems in divergence form.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"197 ","pages":"Pages 19-42"},"PeriodicalIF":2.5,"publicationDate":"2025-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144826942","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":"hp-finite elements for elliptic optimal control problems with control constraints","authors":"Lothar Banz ,&nbsp;Michael Hintermüller ,&nbsp;Andreas Schröder","doi":"10.1016/j.camwa.2025.07.030","DOIUrl":"10.1016/j.camwa.2025.07.030","url":null,"abstract":"<div><div>A distributed elliptic control problem with control constraints is considered, which is formulated as a three field problem and consists of two variational equations for the state and the co-state variables as well as of a variational inequality for the control variable. The adjoint control is associated with the residual of the variational inequality but does not appear in the weak formulation. Each of the three variables is discretized independently by <em>hp</em>-finite elements. In particular, the non-penetration condition of the control variable is relaxed to a finite set of quadrature points. Sufficient conditions for the unique existence of a discrete solution are stated. Also a priori error estimates and guaranteed convergence rates are derived in terms of the mesh size as well as of the polynomial degree. Moreover, reliable and efficient a posteriori error estimates are presented, which enable <em>hp</em>-adaptive mesh refinements. Several numerical experiments demonstrate the applicability of the discretization with <em>hp</em>-finite elements, the efficiency of the a posteriori error estimates and the improvements with respect to the convergence order resulting from the application of <em>hp</em>-adaptivity. In particular, the <em>hp</em>-adaptive schemes lead to superior convergence properties.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"196 ","pages":"Pages 288-311"},"PeriodicalIF":2.5,"publicationDate":"2025-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144771420","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 novel accelerated tempered algorithm with nonuniform time-stepping compact ADI scheme for 2D tempered-fractional nonlinear Schrödinger equations with weak initial singularity","authors":"Himanshu Kumar Dwivedi,&nbsp;Rajeev","doi":"10.1016/j.camwa.2025.07.036","DOIUrl":"10.1016/j.camwa.2025.07.036","url":null,"abstract":"<div><div>This study numerically examines nonlinear fractional Schrödinger equations with the Caputo tempered fractional derivatives (TFD). We present a novel efficient algorithm for enhanced simulation of Caputo TFD. We develop a fast tempered <span><math><mrow><mmultiscripts><mrow><mi>FL</mi></mrow><mprescripts></mprescripts><none></none><mrow><mi>λ</mi></mrow></mmultiscripts></mrow><mn>1</mn></math></span> scheme with parameter <em>λ</em> to significantly reduce computational and storage requirements, crucial for large-scale problems. This algorithm relies on sum of exponents(SOE) technique. The spatial discretization is achieved through the use of the compact finite difference method. Introducing some small perturbation terms yields fully discrete alternating direction implicit (ADI) schemes. By implementing an adaptive time-stepping strategy, we effectively manage long-time simulations while alleviating the inherent initial singularity through the use of a graded mesh in the temporal domain. A crucial Grönwall-type inequality is derived to rigorously analyze the convergence and stability of the proposed <span><math><mrow><mmultiscripts><mrow><mi>FL</mi></mrow><mprescripts></mprescripts><none></none><mrow><mi>λ</mi></mrow></mmultiscripts></mrow><mn>1</mn></math></span>-ADI-CD scheme. Numerical findings are consistent with the anticipated theoretical outcomes, improving precision while substantially lowering computational demands and memory requirements in contrast to classical schemes. This efficiency is further evidenced by a substantial reduction in CPU time. The robustness and reliability of the proposed numerical method are thoroughly validated through extensive numerical experiments. This appears to be first accelerated <span><math><mrow><mmultiscripts><mrow><mi>FL</mi></mrow><mprescripts></mprescripts><none></none><mrow><mi>λ</mi></mrow></mmultiscripts></mrow><mn>1</mn></math></span>-ADI-CD time-stepping approach for nonlinear tempered time fractional Schrödinger equation(NL-TFSE).</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"196 ","pages":"Pages 312-337"},"PeriodicalIF":2.5,"publicationDate":"2025-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144771421","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":"PINN-DTC: Adaptive partitioning of physics-informed neural network for designing thermal cloak","authors":"Ziyu Gan ,&nbsp;Bo Yu","doi":"10.1016/j.camwa.2025.07.037","DOIUrl":"10.1016/j.camwa.2025.07.037","url":null,"abstract":"<div><div>In this study, an adaptive partitioning of physics-informed neural network is proposed for designing thermal cloak (PINN-DTC). In order to realize the intelligent inverse design of thermal cloak structures with different shapes, an adaptive hierarchical inverse design PINN solution framework is established. The application of improved PINN structure for inverse design is somewhat free from the limitation of the background thermal conductivity compared to the thermal cloak research based on the equivalent transformation theory. Consequently, a thermal cloak with considerable functionality can be designed by applying a given material within a certain range. In particular, the presented method avoids the necessity of multiple subdivision and mesh redivision, and simultaneously permits the acquisition of the corresponding full-field temperature and the number of layers of the thermal cloak. Finally, comparing finite element results, the structure of the thermal cloak obtained by PINN-DTC showed superior thermal protection performance.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"195 ","pages":"Pages 419-431"},"PeriodicalIF":2.5,"publicationDate":"2025-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144771525","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":"mGFD: A meshless generalized finite difference method","authors":"Gerardo Tinoco-Guerrero,&nbsp;Francisco Javier Domínguez-Mota,&nbsp;José Alberto Guzmán-Torres,&nbsp;Gabriela Pedraza-Jiménez,&nbsp;José Gerardo Tinoco-Ruiz","doi":"10.1016/j.camwa.2025.07.034","DOIUrl":"10.1016/j.camwa.2025.07.034","url":null,"abstract":"<div><div>This work introduces a novel meshless method, the meshless Generalized Finite Difference (mGFD) scheme, which is derived from an optimization formulation that enforces the consistency condition. This approach eliminates the need for additional weight functions required by other methods, enabling efficient and accurate simulations of complex geometries. The method leverages a flexible, node-based discretization scheme that avoids a predefined mesh, providing enhanced versatility and adaptability in modeling various engineering applications. The proposed method's flexibility and adaptability are demonstrated through numerical solutions of elliptic, parabolic, and hyperbolic partial differential equations in highly irregular domains, providing satisfactory results compared to known exact solutions.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"195 ","pages":"Pages 396-418"},"PeriodicalIF":2.5,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144748781","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":"Isogeometric discretizations of the Stokes problem on trimmed geometries","authors":"Riccardo Puppi","doi":"10.1016/j.camwa.2025.06.032","DOIUrl":"10.1016/j.camwa.2025.06.032","url":null,"abstract":"<div><div>We investigate the isogeometric approximation of the Stokes problem in a trimmed domain, where the underlying mesh is not fitted to the physical domain boundary, posing a challenge for enforcing essential boundary conditions. We introduce three families of isogeometric elements (Raviart-Thomas, Nédélec, and Taylor-Hood) and use them to discretize the problem. The widely used Nitsche method <span><span>[1]</span></span> is commonly employed to address this issue. However, we identify that the Nitsche method lacks stability in certain degenerate trimmed domain configurations, potentially polluting the computed solutions. Our remedy is twofold. On the one hand, we locally change the evaluation of the normal derivatives of the velocities in the weak formulation (generalizing the procedure introduced for the Poisson problem in <span><span>[2]</span></span>); on the other, we modify the space of the discrete pressures, eliminating the degrees of freedom associated with badly trimmed elements. We demonstrate that this approach restores the coercivity of the bilinear form for the velocities. Although numerical results show that our method restores the inf-sup stability of the discrete problem, a rigorous mathematical proof is still missing. We prove optimal a priori error estimates and provide numerical experiments to validate the theory, emphasizing the validation of the inf-sup stability of our method.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"195 ","pages":"Pages 376-395"},"PeriodicalIF":2.5,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144748780","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":"Unconditionally maximum principle preserving scheme for the Allen–Cahn equation with a logarithmic free energy","authors":"Junxiang Yang ,&nbsp;Sangkwon Kim ,&nbsp;Junseok Kim","doi":"10.1016/j.camwa.2025.07.032","DOIUrl":"10.1016/j.camwa.2025.07.032","url":null,"abstract":"<div><div>We present a novel computational scheme that unconditionally satisfies the maximum principle to solve the Allen–Cahn (AC) equation with a logarithmic Flory–Huggins potential. The proposed scheme uses an operator splitting approach integrated with a frozen coefficient technique without using any stabilization term. Due to the use of the frozen coefficient method, we can apply a closed-form solution to the highly nonlinear term. This combination allows the development of a scheme that rigorously maintains the maximum principle throughout the computation. We provide a detailed analytical proof of the discrete maximum principle for the proposed scheme to ensure its theoretical robustness. To validate the high performance, we conduct computational experiments, which confirm the unconditional stability and demonstrate the method's effectiveness in preserving the maximum principle. These numerical results highlight the proposed scheme's potential as a reliable tool for accurately solving the AC equation.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"195 ","pages":"Pages 366-375"},"PeriodicalIF":2.5,"publicationDate":"2025-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144721592","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":"Formulae and transformations for simplicial tensorial finite elements via polytopal templates","authors":"Adam Sky ,&nbsp;Michael Neunteufel ,&nbsp;Jack S. Hale ,&nbsp;Andreas Zilian","doi":"10.1016/j.camwa.2025.07.028","DOIUrl":"10.1016/j.camwa.2025.07.028","url":null,"abstract":"<div><div>We introduce a unified method for constructing the basis functions of a wide variety of partially continuous tensor-valued finite elements on simplices using polytopal templates. These finite element spaces are essential for achieving well-posed discretisations of mixed formulations of partial differential equations that involve tensor-valued functions, such as the Hellinger–Reissner formulation of linear elasticity. In our proposed polytopal template method, the basis functions are constructed from template tensors associated with the geometric polytopes (vertices, edges, faces etc.) of the reference simplex and any scalar-valued <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-conforming finite element space. From this starting point we can construct the Regge, Hellan–Herrmann–Johnson, Pechstein–Schöberl, Hu–Zhang, Hu–Ma–Sun and Gopalakrishnan–Lederer–Schöberl elements. Because the Hu–Zhang element and the Hu–Ma–Sun element cannot be mapped from the reference simplex to a physical simplex via standard double Piola mappings, we also demonstrate that the polytopal template tensors can be used to define a consistent mapping from a reference simplex even to a non-affine simplex in the physical mesh. Finally, we discuss the implications of element regularity with two numerical examples for the Reissner–Mindlin plate problem.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"195 ","pages":"Pages 322-348"},"PeriodicalIF":2.9,"publicationDate":"2025-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144713999","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 a discontinuous Galerkin method for time-fractional Navier-Stokes-Fokker-Planck equations with weakly singular solutions","authors":"Dong Liu ,&nbsp;Weihua Deng","doi":"10.1016/j.camwa.2025.07.031","DOIUrl":"10.1016/j.camwa.2025.07.031","url":null,"abstract":"<div><div>In this paper, a class of time-fractional Navier-Stokes-Fokker-Planck equations (TF-NSFPEs) describing position of particles with anomalous diffusion in viscous incompressible fluids are proposed. We develop the symmetric interior penalty discontinuous Galerkin (IPDG) method for TF-NSFPEs. The <em>L</em>1 method in the time on graded mesh is used for the reason that the solution of the time fractional Fokker-Planck equation (TFFPE) usually has a weak singularity near the initial time. The stability and the optimal error estimates of the IPDG semi-discrete scheme are proved by using the discrete fractional Grönwall inequality. Further, based on these results, the fully discrete optimal error estimates are obtained. Finally, some numerical experiments are performed to justify the effectiveness of the theoretical results.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"195 ","pages":"Pages 280-295"},"PeriodicalIF":2.9,"publicationDate":"2025-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144713797","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}