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A new primal-dual hybrid gradient scheme for solving minimax problems with nonlinear term
IF 2.2 2区 数学
Applied Numerical Mathematics Pub Date : 2024-12-27 DOI: 10.1016/j.apnum.2024.12.010
Renkai Wu, Zexian Liu
{"title":"A new primal-dual hybrid gradient scheme for solving minimax problems with nonlinear term","authors":"Renkai Wu,&nbsp;Zexian Liu","doi":"10.1016/j.apnum.2024.12.010","DOIUrl":"10.1016/j.apnum.2024.12.010","url":null,"abstract":"<div><div>Primal-dual hybrid gradient (PDHG) methods are popular for solving minimax problems. The proximal terms in the corresponding subproblems play an important role in the convergence analysis and for numerical performance of PDHG methods. However, it is observed that the function values generated by some PDHG algorithms might suffer from intense oscillation as the iteration progresses. To address the drawback, we take advantage of an inertial point to exploit a new proximal term, construct a new quadratic approximation for the nonlinear term in the minimax problem, and present a new primal-dual hybrid gradient algorithm for solving minimax problems with nonlinear terms. The new proximal term is different from other commonly used proximal terms and is used in the <em>x</em>-subproblem of the proposed algorithm. The quadratic approximation is used to replace the common linear approximation in the subproblem of the proposed algorithm to accelerate the proposed method. The local convergence of the proposed algorithm is established under mild assumptions. Numerical experiments on two examples confirm the compelling numerical performance of the proposed method.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"210 ","pages":"Pages 147-164"},"PeriodicalIF":2.2,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143175059","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}
引用次数: 0
Strongly consistent low-dissipation WENO schemes for finite elements
IF 2.2 2区 数学
Applied Numerical Mathematics Pub Date : 2024-12-18 DOI: 10.1016/j.apnum.2024.12.008
Joshua Vedral , Andreas Rupp , Dmitri Kuzmin
{"title":"Strongly consistent low-dissipation WENO schemes for finite elements","authors":"Joshua Vedral ,&nbsp;Andreas Rupp ,&nbsp;Dmitri Kuzmin","doi":"10.1016/j.apnum.2024.12.008","DOIUrl":"10.1016/j.apnum.2024.12.008","url":null,"abstract":"<div><div>We propose a way to maintain strong consistency and perform error analysis in the context of dissipation-based WENO stabilization for continuous and discontinuous Galerkin discretizations of conservation laws. Following Kuzmin and Vedral (J. Comput. Phys. 487:112153, 2023) and Vedral (arXiv preprint <span><span>arXiv:2309.12019</span><svg><path></path></svg></span>), we use WENO shock detectors to determine appropriate amounts of low-order artificial viscosity. In contrast to existing WENO methods, our approach blends candidate polynomials using residual-based nonlinear weights. The shock-capturing terms of our stabilized Galerkin methods vanish if residuals do. This enables us to achieve improved accuracy compared to weakly consistent alternatives. As we show in the context of steady convection-diffusion-reaction (CDR) equations, nonlinear local projection stabilization terms can be included in a way that preserves the coercivity of local bilinear forms. For the corresponding Galerkin-WENO discretization of a CDR problem, we rigorously derive a priori error estimates. Additionally, we demonstrate the stability and accuracy of the proposed method through one- and two-dimensional numerical experiments for hyperbolic conservation laws and systems thereof. The numerical results for representative test problems are superior to those obtained with traditional WENO schemes, particularly in scenarios involving shocks and steep gradients.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"210 ","pages":"Pages 64-81"},"PeriodicalIF":2.2,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143175013","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Commutator-based operator splitting for linear port-Hamiltonian systems
IF 2.2 2区 数学
Applied Numerical Mathematics Pub Date : 2024-12-16 DOI: 10.1016/j.apnum.2024.12.007
Marius Mönch, Nicole Marheineke
{"title":"Commutator-based operator splitting for linear port-Hamiltonian systems","authors":"Marius Mönch,&nbsp;Nicole Marheineke","doi":"10.1016/j.apnum.2024.12.007","DOIUrl":"10.1016/j.apnum.2024.12.007","url":null,"abstract":"<div><div>In this paper, we develop high-order splitting methods for linear port-Hamiltonian systems, focusing on preserving their intrinsic structure, particularly the dissipation inequality. Port-Hamiltonian systems are characterized by their ability to describe energy-conserving and dissipative processes, which is essential for the accurate simulation of physical systems. For autonomous systems, we introduce an energy-associated decomposition that exploits the system's energy properties. We present splitting schemes up to order six. In the non-autonomous case, we employ a port-based splitting. This special technique makes it possible to set up methods of arbitrary even order. Both splitting approaches are based on the properties of the commutator and ensure that the numerical schemes not only preserve the structure of the system but also faithfully fulfill the dissipation inequality. The proposed approaches are validated through theoretical analysis and numerical experiments.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"210 ","pages":"Pages 25-38"},"PeriodicalIF":2.2,"publicationDate":"2024-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143175080","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The two-grid hybrid high-order method for the nonlinear strongly damped wave equation on polygonal mesh and its reduced-order model
IF 2.2 2区 数学
Applied Numerical Mathematics Pub Date : 2024-12-16 DOI: 10.1016/j.apnum.2024.12.006
Lu Wang, Youjun Tan, Minfu Feng
{"title":"The two-grid hybrid high-order method for the nonlinear strongly damped wave equation on polygonal mesh and its reduced-order model","authors":"Lu Wang,&nbsp;Youjun Tan,&nbsp;Minfu Feng","doi":"10.1016/j.apnum.2024.12.006","DOIUrl":"10.1016/j.apnum.2024.12.006","url":null,"abstract":"<div><div>This paper introduces the hybrid high-order (HHO) method for solving the nonlinear strongly damped wave equation. We comprehensively analyze the semi-discrete and fully-discrete implicit schemes, including energy and <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> norm, with convergence rates of <span><math><mi>m</mi><mo>+</mo><mn>1</mn></math></span> and <span><math><mi>m</mi><mo>+</mo><mn>2</mn></math></span> in space (<span><math><mi>m</mi><mo>≥</mo><mn>0</mn></math></span>), respectively. In addition, we combine the two-grid algorithm (TGA) with the HHO method (TGA-HHO) to improve computational efficiency and analyze the TGA-HHO method. To improve the computational efficiency further, we combine the proper orthogonal decomposition (POD) technique with the TGA-HHO method (POD-TGA-HHO). Finally, we provide numerical examples to validate the effectiveness of the HHO, TGA-HHO, and POD-TGA-HHO algorithms.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"210 ","pages":"Pages 1-24"},"PeriodicalIF":2.2,"publicationDate":"2024-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143175082","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}
引用次数: 0
A novel efficient generalized energy-optimized exponential SAV scheme with variable-step BDFk method for gradient flows
IF 2.2 2区 数学
Applied Numerical Mathematics Pub Date : 2024-12-14 DOI: 10.1016/j.apnum.2024.12.005
Bingyin Zhang , Chengxi Zhou , Hongfei Fu
{"title":"A novel efficient generalized energy-optimized exponential SAV scheme with variable-step BDFk method for gradient flows","authors":"Bingyin Zhang ,&nbsp;Chengxi Zhou ,&nbsp;Hongfei Fu","doi":"10.1016/j.apnum.2024.12.005","DOIUrl":"10.1016/j.apnum.2024.12.005","url":null,"abstract":"<div><div>In this paper, we propose a novel generalized energy-optimized (GEOP) technique to correct the modified energy of the scalar auxiliary variable (SAV) approach for gradient flows. Firstly, we use the variable-step <em>k</em>th-order (<em>k</em> = 2,3) backward differentiation formula (BDF<em>k</em>) to construct a linear exponential SAV method (ESAV), denoted as BDF<em>k</em>-ESAV. This method is shown to preserve only a modified energy dissipation law. To address this issue, we present an energy-optimized (EOP) technique derived from a novel linear relaxation strategy, which penalizes the inconsistency between the SAV and the original nonlinear potential energy. However, this does not always bring the modified energy closer to the original total energy. This paper presents one essential GEOP technique to overcome this issue, which leads to a novel ESAV scheme, namely BDF<em>k</em>-GEOP-ESAV. We demonstrate that this scheme unconditionally satisfies the modified energy dissipation law, similar to the proposed ESAV and EOP-ESAV schemes. Most importantly, its energy is an optimal approximation to the original total energy, not just the nonlinear potential energy. Therefore, it enables a broad range of applications for long-term stable modeling. Additionally, an improved adaptive time-stepping strategy is developed to further enhance the effectiveness and efficiency of the variable-step BDF<em>k</em>-GEOP-ESAV method. Representative numerical examples are presented to illustrate the superior performance of the proposed methods.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"210 ","pages":"Pages 39-63"},"PeriodicalIF":2.2,"publicationDate":"2024-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143175079","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}
引用次数: 0
The linear, decoupled and fully discrete finite element methods for simplified two-phase ferrohydrodynamics model
IF 2.2 2区 数学
Applied Numerical Mathematics Pub Date : 2024-12-14 DOI: 10.1016/j.apnum.2024.12.004
Xiaoyong Chen , Rui Li , Jian Li
{"title":"The linear, decoupled and fully discrete finite element methods for simplified two-phase ferrohydrodynamics model","authors":"Xiaoyong Chen ,&nbsp;Rui Li ,&nbsp;Jian Li","doi":"10.1016/j.apnum.2024.12.004","DOIUrl":"10.1016/j.apnum.2024.12.004","url":null,"abstract":"<div><div>In this paper, we consider numerical approximations of a phase field model for simplified two-phase ferrofluids. This model is a highly nonlinear and coupled multiphysics PDE system with Cahn-Hilliard equations, Navier-Stokes equations, magnetization equation and magnetostatic equation. By combining the artificial compressibility method for the Navier-Stokes equations, the convex splitting method or the stabilize explicit method for Cahn-Hilliard systems, the subtle implicit-explicit treatments and some extra stabilization terms for nonlinear coupling terms, we construct two linear, decoupled and fully discrete finite element methods to solve multiphysics system efficiently. The proposed schemes do not enforce any artificial boundary condition on the pressure. Furthermore, the energy stability and unique solvability are obtained for the proposed schemes. In order to accurately capture the diffuse interface, we apply the adaptive mesh strategy. Finally, a series of numerical experiments verify the theory and illustrate the efficiency and effectiveness of these methods.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"210 ","pages":"Pages 123-146"},"PeriodicalIF":2.2,"publicationDate":"2024-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143175060","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}
引用次数: 0
Special volume on Advanced Mathematical and Numerical Models in Applied Sciences (AMNMAS)
IF 2.2 2区 数学
Applied Numerical Mathematics Pub Date : 2024-12-11 DOI: 10.1016/j.apnum.2024.12.003
Elisa Francomano (Managing Guest Editor) , Stefano De Marchi (Guest Editor) , Galina Filipuk (Guest Editor) , Giuliana Ramella (Guest Editor) , Federico Zullo (Guest Editor)
{"title":"Special volume on Advanced Mathematical and Numerical Models in Applied Sciences (AMNMAS)","authors":"Elisa Francomano (Managing Guest Editor) ,&nbsp;Stefano De Marchi (Guest Editor) ,&nbsp;Galina Filipuk (Guest Editor) ,&nbsp;Giuliana Ramella (Guest Editor) ,&nbsp;Federico Zullo (Guest Editor)","doi":"10.1016/j.apnum.2024.12.003","DOIUrl":"10.1016/j.apnum.2024.12.003","url":null,"abstract":"","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"210 ","pages":"Page 82"},"PeriodicalIF":2.2,"publicationDate":"2024-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143175055","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}
引用次数: 0
Boundary corrections for splitting methods in the time integration of multidimensional parabolic problems
IF 2.2 2区 数学
Applied Numerical Mathematics Pub Date : 2024-12-06 DOI: 10.1016/j.apnum.2024.12.002
S. González-Pinto, D. Hernández-Abreu
{"title":"Boundary corrections for splitting methods in the time integration of multidimensional parabolic problems","authors":"S. González-Pinto,&nbsp;D. Hernández-Abreu","doi":"10.1016/j.apnum.2024.12.002","DOIUrl":"10.1016/j.apnum.2024.12.002","url":null,"abstract":"<div><div>This work considers two boundary correction techniques to mitigate the reduction in the temporal order of convergence in PDE sense (i.e., when both the space and time resolutions tend to zero independently of each other) of <em>d</em> dimension space-discretized parabolic problems on a rectangular domain subject to time dependent boundary conditions. We make use of the MoL approach (method of lines) where the space discretization is made with central differences of order four and the time integration is carried out with <em>s</em>-stage AMF-W-methods. The time integrators are of ADI-type (alternating direction implicit by using a directional splitting) and of higher order than the usual ones appearing in the literature which only reach order two. Besides, the techniques here explained also work for most of splitting methods when directional splitting is used. A remarkable fact is that with these techniques the time integrators recover the temporal order of PDE-convergence at the level of time-independent boundary conditions.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"210 ","pages":"Pages 95-112"},"PeriodicalIF":2.2,"publicationDate":"2024-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143175066","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A numerical approach for a 1D tumor-angiogenesis simulations model
IF 2.2 2区 数学
Applied Numerical Mathematics Pub Date : 2024-12-06 DOI: 10.1016/j.apnum.2024.11.017
P. De Luca , A. Galletti , G. Giunta , L. Marcellino
{"title":"A numerical approach for a 1D tumor-angiogenesis simulations model","authors":"P. De Luca ,&nbsp;A. Galletti ,&nbsp;G. Giunta ,&nbsp;L. Marcellino","doi":"10.1016/j.apnum.2024.11.017","DOIUrl":"10.1016/j.apnum.2024.11.017","url":null,"abstract":"<div><div>Angiogenesis, the formation of new blood vessels, is critical in both normal and pathological contexts, especially cancer. This process involves complex interactions among endothelial cells, tumor angiogenic factors, matrix metalloproteinases, angiogenic inhibitors, and neoplastic tissues. Different mathematical and computational models have been proposed for representing the tumor angiogenesis process. Among these, we focus on partial differential equations models which are able to capture the dynamic and spatial complexities in tumor growing. Our starting point is a PDE system which mimics the angiogenesis evolution. The aim of this work is to combine both spatial and time discretization methods for designing a matrix-based model. This approach allows us to observe some error properties of numerical schema proposed, by deducing the cumulative error among space and time. Experimental tests include convergence studies, for validating the reliability of the method. Results confirm our approach is useful for addressing real angiogenesis problem.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"210 ","pages":"Pages 83-94"},"PeriodicalIF":2.2,"publicationDate":"2024-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143175065","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}
引用次数: 0
Error analysis of an ADI scheme for the two-dimensional fractal mobile/immobile transport equation with weakly singular solutions
IF 2.2 2区 数学
Applied Numerical Mathematics Pub Date : 2024-12-05 DOI: 10.1016/j.apnum.2024.12.001
Weizhi Liu , Hu Chen , Mahmoud Zaky
{"title":"Error analysis of an ADI scheme for the two-dimensional fractal mobile/immobile transport equation with weakly singular solutions","authors":"Weizhi Liu ,&nbsp;Hu Chen ,&nbsp;Mahmoud Zaky","doi":"10.1016/j.apnum.2024.12.001","DOIUrl":"10.1016/j.apnum.2024.12.001","url":null,"abstract":"<div><div>In this work, we consider a numerical approximation for the two-dimensional fractal mobile/immobile transport equation with weakly singular solutions, where the time first-order derivative is discretized by the backward Euler method, and the Caputo fractional derivative is approximated by the L1 scheme on a uniform mesh. The fully discrete ADI scheme is established by adding a high-order term. The stability and the convergence analyses of the fully discrete ADI scheme are analyzed in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norm and <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-norm. The numerical results show that the error estimates are sharp.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"210 ","pages":"Pages 113-122"},"PeriodicalIF":2.2,"publicationDate":"2024-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143175056","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}
引用次数: 0
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