SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1886-1908, August 2025. Abstract. The parallel orbital-updating approach is an orbital/eigenfunction iteration based approach for solving eigenvalue problems when many eigenpairs are required. It has been proven to be efficient, for instance, in electronic structure calculations. In this paper, based on the investigation of a quasi-orthogonality, we present the numerical analysis of the parallel orbital-updating approach for linear eigenvalue problems, including convergence and error estimates of the numerical approximations.
{"title":"Numerical Analysis of the Parallel Orbital-Updating Approach for Eigenvalue Problems","authors":"Xiaoying Dai, Yan Li, Bin Yang, Aihui Zhou","doi":"10.1137/24m1690084","DOIUrl":"https://doi.org/10.1137/24m1690084","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1886-1908, August 2025. <br/> Abstract. The parallel orbital-updating approach is an orbital/eigenfunction iteration based approach for solving eigenvalue problems when many eigenpairs are required. It has been proven to be efficient, for instance, in electronic structure calculations. In this paper, based on the investigation of a quasi-orthogonality, we present the numerical analysis of the parallel orbital-updating approach for linear eigenvalue problems, including convergence and error estimates of the numerical approximations.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"15 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144900116","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}
SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1808-1832, August 2025. Abstract. The main theoretical obstacle to establishing the original energy dissipation laws of Runge–Kutta methods for phase field equations is verifying the maximum norm boundedness of the stage solutions without assuming global Lipschitz continuity of the nonlinear bulk. We present a unified theoretical framework for the energy stability of three effective classes of Runge–Kutta methods, including the additive implicit-explicit Runge–Kutta, explicit exponential Runge–Kutta, and corrected integrating factor Runge–Kutta methods, for the Swift–Hohenberg and phase field crystal models. By the standard discrete energy argument, it is proven that the three classes of Runge–Kutta methods preserve the original energy dissipation law if the associated differentiation matrices are positive definite. Our main tools include the differential form with the associated differentiation matrix, the discrete orthogonal convolution kernel, and the principle of mathematical induction. Many existing Runge–Kutta methods in the literature are revisited by evaluating the lower bound on the minimum eigenvalues of the associated differentiation matrices. Our theoretical approach paves a new way toward the internal nonlinear stability of Runge–Kutta methods for dissipative semilinear parabolic problems.
{"title":"A Unified Framework on the Original Energy Laws of Three Effective Classes of Runge–Kutta Methods for Phase Field Crystal Type Models","authors":"Xuping Wang, Xuan Zhao, Hong-lin Liao","doi":"10.1137/24m1701770","DOIUrl":"https://doi.org/10.1137/24m1701770","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1808-1832, August 2025. <br/> Abstract. The main theoretical obstacle to establishing the original energy dissipation laws of Runge–Kutta methods for phase field equations is verifying the maximum norm boundedness of the stage solutions without assuming global Lipschitz continuity of the nonlinear bulk. We present a unified theoretical framework for the energy stability of three effective classes of Runge–Kutta methods, including the additive implicit-explicit Runge–Kutta, explicit exponential Runge–Kutta, and corrected integrating factor Runge–Kutta methods, for the Swift–Hohenberg and phase field crystal models. By the standard discrete energy argument, it is proven that the three classes of Runge–Kutta methods preserve the original energy dissipation law if the associated differentiation matrices are positive definite. Our main tools include the differential form with the associated differentiation matrix, the discrete orthogonal convolution kernel, and the principle of mathematical induction. Many existing Runge–Kutta methods in the literature are revisited by evaluating the lower bound on the minimum eigenvalues of the associated differentiation matrices. Our theoretical approach paves a new way toward the internal nonlinear stability of Runge–Kutta methods for dissipative semilinear parabolic problems.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"15 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144900120","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}
SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1729-1756, August 2025. Abstract. We consider a novel way of discretizing wave scattering problems using the general formalism of convolution quadrature, but instead of reducing the time step size ([math]-method), we achieve accuracy by increasing the order of the method ([math]-method). We base this method on discontinuous Galerkin time stepping and use the Z-transform. We show that for a certain class of incident waves, the resulting schemes observe a (root)-exponential convergence rate with respect to the number of boundary integral operators that need to be applied. Numerical experiments confirm the finding.
{"title":"A P-Version of Convolution Quadrature in Wave Propagation","authors":"Alexander Rieder","doi":"10.1137/24m1642524","DOIUrl":"https://doi.org/10.1137/24m1642524","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1729-1756, August 2025. <br/> Abstract. We consider a novel way of discretizing wave scattering problems using the general formalism of convolution quadrature, but instead of reducing the time step size ([math]-method), we achieve accuracy by increasing the order of the method ([math]-method). We base this method on discontinuous Galerkin time stepping and use the Z-transform. We show that for a certain class of incident waves, the resulting schemes observe a (root)-exponential convergence rate with respect to the number of boundary integral operators that need to be applied. Numerical experiments confirm the finding.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"105 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144840295","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}
Hua Su, Haoran Wang, Lei Zhang, Jin Zhao, Xiangcheng Zheng
SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1757-1775, August 2025. Abstract. We present an improved high-index saddle dynamics (iHiSD) for finding saddle points and constructing solution landscapes, which is a crossover dynamics from gradient flow to traditional HiSD such that the Morse theory for gradient flow could be involved. We propose analysis for the reflection manifold in iHiSD and then prove its stable and nonlocal convergence from stationary points that may not be close to the target saddle point, which reduces the dependence of the convergence of HiSD on the initial value. We then present and analyze a discretized iHiSD for implementation. Furthermore, based on Morse theory, we prove that any two saddle points could be connected by a sequence of trajectories of iHiSD. Ideally, this implies that a solution landscape with a finite number of stationary points could be completely constructed by means of iHiSD, which partly answers the completeness issue of the solution landscape for the first time and indicates the necessity of integrating the gradient flow in HiSD. Different methods are compared by numerical experiments to substantiate the effectiveness of the iHiSD method.
{"title":"Improved High-Index Saddle Dynamics for Finding Saddle Points and Solution Landscape","authors":"Hua Su, Haoran Wang, Lei Zhang, Jin Zhao, Xiangcheng Zheng","doi":"10.1137/25m173212x","DOIUrl":"https://doi.org/10.1137/25m173212x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1757-1775, August 2025. <br/> Abstract. We present an improved high-index saddle dynamics (iHiSD) for finding saddle points and constructing solution landscapes, which is a crossover dynamics from gradient flow to traditional HiSD such that the Morse theory for gradient flow could be involved. We propose analysis for the reflection manifold in iHiSD and then prove its stable and nonlocal convergence from stationary points that may not be close to the target saddle point, which reduces the dependence of the convergence of HiSD on the initial value. We then present and analyze a discretized iHiSD for implementation. Furthermore, based on Morse theory, we prove that any two saddle points could be connected by a sequence of trajectories of iHiSD. Ideally, this implies that a solution landscape with a finite number of stationary points could be completely constructed by means of iHiSD, which partly answers the completeness issue of the solution landscape for the first time and indicates the necessity of integrating the gradient flow in HiSD. Different methods are compared by numerical experiments to substantiate the effectiveness of the iHiSD method.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"1 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144840296","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}
SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1691-1728, August 2025. Abstract. In this article, we propose and study a stochastic and relaxed preconditioned Douglas–Rachford splitting method to solve saddle-point problems that have separable dual variables. We prove the almost sure convergence of the iteration sequences in Hilbert spaces for a class of convex-concave and nonsmooth saddle-point problems. We also provide the sublinear convergence rate for the ergodic sequence concerning the expectation of the restricted primal-dual gap functions. Numerical experiments show the high efficiency of the proposed stochastic and relaxed preconditioned Douglas–Rachford splitting methods.
{"title":"A Stochastic Preconditioned Douglas–Rachford Splitting Method for Saddle-Point Problems","authors":"Yakun Dong, Kristian Bredies, Hongpeng Sun","doi":"10.1137/23m1622490","DOIUrl":"https://doi.org/10.1137/23m1622490","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1691-1728, August 2025. <br/> Abstract. In this article, we propose and study a stochastic and relaxed preconditioned Douglas–Rachford splitting method to solve saddle-point problems that have separable dual variables. We prove the almost sure convergence of the iteration sequences in Hilbert spaces for a class of convex-concave and nonsmooth saddle-point problems. We also provide the sublinear convergence rate for the ergodic sequence concerning the expectation of the restricted primal-dual gap functions. Numerical experiments show the high efficiency of the proposed stochastic and relaxed preconditioned Douglas–Rachford splitting methods.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"15 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144819998","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}
SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1666-1690, August 2025. Abstract. There has been a surge of interest in uncertainty quantification for parametric partial differential equations (PDEs) with Gevrey regular inputs. The Gevrey class contains functions that are infinitely smooth with a growth condition on the higher-order partial derivatives, but which are nonetheless not analytic in general. Recent studies by Chernov and Lê [Comput. Math. Appl., 164 (2024), pp. 116–130; SIAM J. Numer. Anal., 62 (2024), pp. 1874–1900] as well as Harbrecht, Schmidlin, and Schwab [Math. Models Methods Appl. Sci., 34 (2024), pp. 881–917] analyze the setting wherein the input random field is assumed to be uniformly bounded with respect to the uncertain parameters. In this paper, we relax this assumption and allow for parameter-dependent bounds. The parametric inputs are modeled as generalized Gaussian random variables, and we analyze the application of quasi-Monte Carlo (QMC) integration to assess the PDE response statistics using randomly shifted rank-1 lattice rules. In addition to the QMC error analysis, we also consider the dimension truncation and finite element errors in this setting.
SIAM数值分析杂志，第63卷，第4期，1666-1690页，2025年8月。摘要。对具有格弗雷正则输入的参数偏微分方程（PDEs）的不确定性量化的兴趣激增。Gevrey类包含具有高阶偏导数生长条件的无限光滑函数，但通常不是解析函数。Chernov和Lê最近的研究[Comput。数学。达成。， 164(2024)，第116-130页；SIAM J. number。分析的。， 62 (2024), pp. 1874-1900]以及Harbrecht， Schmidlin和Schwab[数学。模型、方法、应用。科学。[j], 34 (2024), pp. 881-917]分析假设输入随机场相对于不确定参数是均匀有界的设置。在本文中，我们放宽了这个假设，并允许参数相关的边界。将参数输入建模为广义高斯随机变量，并分析了拟蒙特卡罗积分（QMC）的应用，利用随机移位秩-1格规则来评估PDE响应统计量。除了QMC误差分析外，我们还考虑了尺寸截断和有限元误差。
{"title":"Quasi-Monte Carlo for Partial Differential Equations with Generalized Gaussian Input Uncertainty","authors":"Philipp A. Guth, Vesa Kaarnioja","doi":"10.1137/24m1708164","DOIUrl":"https://doi.org/10.1137/24m1708164","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1666-1690, August 2025. <br/> Abstract. There has been a surge of interest in uncertainty quantification for parametric partial differential equations (PDEs) with Gevrey regular inputs. The Gevrey class contains functions that are infinitely smooth with a growth condition on the higher-order partial derivatives, but which are nonetheless not analytic in general. Recent studies by Chernov and Lê [Comput. Math. Appl., 164 (2024), pp. 116–130; SIAM J. Numer. Anal., 62 (2024), pp. 1874–1900] as well as Harbrecht, Schmidlin, and Schwab [Math. Models Methods Appl. Sci., 34 (2024), pp. 881–917] analyze the setting wherein the input random field is assumed to be uniformly bounded with respect to the uncertain parameters. In this paper, we relax this assumption and allow for parameter-dependent bounds. The parametric inputs are modeled as generalized Gaussian random variables, and we analyze the application of quasi-Monte Carlo (QMC) integration to assess the PDE response statistics using randomly shifted rank-1 lattice rules. In addition to the QMC error analysis, we also consider the dimension truncation and finite element errors in this setting.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"42 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144819942","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}
SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1642-1665, August 2025. Abstract. This paper presents a novel stabilized nonconforming finite element method for solving the surface biharmonic problem. The method extends the New-Zienkiewicz-type (NZT) element to polyhedral (approximated) surfaces by employing the Piola transform to establish the connection of vertex gradients across adjacent elements. Key features of the surface NZT finite element space include its [math]-relative conformity and weak [math] conformity, allowing for stabilization without the use of artificial parameters. Under the assumption that the exact solution and the dual problem possess only [math] regularity, we establish optimal error estimates in the energy norm and provide, for the first time, a comprehensive analysis yielding optimal second-order convergence in the broken [math] norm. Numerical experiments are provided to support the theoretical results and indicate that the stabilization term might be unnecessary.
{"title":"A Stabilized Nonconforming Finite Element Method for the Surface Biharmonic Problem","authors":"Shuonan Wu, Hao Zhou","doi":"10.1137/24m1707936","DOIUrl":"https://doi.org/10.1137/24m1707936","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1642-1665, August 2025. <br/> Abstract. This paper presents a novel stabilized nonconforming finite element method for solving the surface biharmonic problem. The method extends the New-Zienkiewicz-type (NZT) element to polyhedral (approximated) surfaces by employing the Piola transform to establish the connection of vertex gradients across adjacent elements. Key features of the surface NZT finite element space include its [math]-relative conformity and weak [math] conformity, allowing for stabilization without the use of artificial parameters. Under the assumption that the exact solution and the dual problem possess only [math] regularity, we establish optimal error estimates in the energy norm and provide, for the first time, a comprehensive analysis yielding optimal second-order convergence in the broken [math] norm. Numerical experiments are provided to support the theoretical results and indicate that the stabilization term might be unnecessary.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"16 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144787661","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}
SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1617-1641, August 2025. Abstract. In this paper, we propose a multiscale method for heterogeneous Stokes problems. The method is based on the localized orthogonal decomposition (LOD) methodology and has approximation properties independent of the regularity of the coefficients. We apply the LOD to an appropriate reformulation of the Stokes problem, which allows us to construct exponentially decaying basis functions for the velocity approximation while using a piecewise constant pressure approximation. The exponential decay motivates a localization of the basis computation, which is essential for the practical realization of the method. We perform a rigorous a priori error analysis and prove optimal convergence rates for the velocity approximation and a postprocessed pressure approximation, provided that the supports of the basis functions are logarithmically increased with the desired accuracy. Numerical experiments support the theoretical results of this paper.
{"title":"A Localized Orthogonal Decomposition Method for Heterogeneous Stokes Problems","authors":"Moritz Hauck, Alexei Lozinski","doi":"10.1137/24m1704166","DOIUrl":"https://doi.org/10.1137/24m1704166","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1617-1641, August 2025. <br/> Abstract. In this paper, we propose a multiscale method for heterogeneous Stokes problems. The method is based on the localized orthogonal decomposition (LOD) methodology and has approximation properties independent of the regularity of the coefficients. We apply the LOD to an appropriate reformulation of the Stokes problem, which allows us to construct exponentially decaying basis functions for the velocity approximation while using a piecewise constant pressure approximation. The exponential decay motivates a localization of the basis computation, which is essential for the practical realization of the method. We perform a rigorous a priori error analysis and prove optimal convergence rates for the velocity approximation and a postprocessed pressure approximation, provided that the supports of the basis functions are logarithmically increased with the desired accuracy. Numerical experiments support the theoretical results of this paper.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"69 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144766090","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}
SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1586-1616, August 2025. Abstract. We present a new stability and error analysis of fully discrete approximation schemes for the transient Stokes equation. For the spatial discretization, we consider a wide class of Galerkin finite element methods which includes both inf-sup stable spaces and symmetric pressure stabilized formulations. We extend the results from Burman and Fernández [SIAM J. Numer. Anal., 47 (2009), pp. 409–439] and provide a unified theoretical analysis of backward difference formula methods of orders 1 to 6. The main novelty of our approach lies in deriving optimal-order stability and error estimates for both the velocity and the pressure using Dahlquist’s [math]-stability concept together with the multiplier technique introduced by Nevanlinna and Odeh and recently by Akrivis et al. [SIAM J. Numer. Anal., 59 (2021), pp. 2449–2472]. When combined with a method-dependent Ritz projection for the initial data, unconditional stability can be shown, while for arbitrary interpolation, pressure stability is subordinate to the fulfillment of a mild inverse CFL-type condition between space and time discretizations.
SIAM数值分析杂志，第63卷，第4期，第1586-1616页，2025年8月。摘要。给出了暂态Stokes方程全离散近似格式的一种新的稳定性和误差分析。对于空间离散化，我们考虑了一种广泛的Galerkin有限元方法，它既包括中支撑稳定空间，也包括对称压力稳定公式。我们扩展了Burman和Fernández [SIAM J. number]的结果。分析的。， 47 (2009), pp. 409-439]，并对1 ~ 6阶的后向差分公式方法进行了统一的理论分析。该方法的主要新颖之处在于利用Dahlquist的[数学]稳定性概念以及Nevanlinna和Odeh以及最近由Akrivis等人引入的乘法器技术，推导出速度和压力的最优阶稳定性和误差估计。分析的。， 59 (2021), pp. 2449-2472]。当与初始数据的方法相关的Ritz投影相结合时，可以显示出无条件的稳定性，而对于任意插值，压力稳定性从属于满足空间和时间离散之间的温和逆cfl型条件。
{"title":"Error Analysis of BDF 1–6 Time-Stepping Methods for the Transient Stokes Problem: Velocity and Pressure Estimates","authors":"Alessandro Contri, Balázs Kovács, André Massing","doi":"10.1137/23m1606800","DOIUrl":"https://doi.org/10.1137/23m1606800","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1586-1616, August 2025. <br/> Abstract. We present a new stability and error analysis of fully discrete approximation schemes for the transient Stokes equation. For the spatial discretization, we consider a wide class of Galerkin finite element methods which includes both inf-sup stable spaces and symmetric pressure stabilized formulations. We extend the results from Burman and Fernández [SIAM J. Numer. Anal., 47 (2009), pp. 409–439] and provide a unified theoretical analysis of backward difference formula methods of orders 1 to 6. The main novelty of our approach lies in deriving optimal-order stability and error estimates for both the velocity and the pressure using Dahlquist’s [math]-stability concept together with the multiplier technique introduced by Nevanlinna and Odeh and recently by Akrivis et al. [SIAM J. Numer. Anal., 59 (2021), pp. 2449–2472]. When combined with a method-dependent Ritz projection for the initial data, unconditional stability can be shown, while for arbitrary interpolation, pressure stability is subordinate to the fulfillment of a mild inverse CFL-type condition between space and time discretizations.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"119 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144702055","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}
SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1561-1585, August 2025. Abstract. We propose a modified Trefftz discontinuous Galerkin (TDG) method for approximating a time-harmonic acoustic scattering problem in an infinitely elongated waveguide. In the waveguide we suppose that there is a bounded, penetrable, and possibly absorbing scatterer. The classical TDG is not applicable when the scatterer is absorbing. Novel features of our modified TDG method are that it is applicable in this case, and it uses a stable treatment of the outgoing radiation condition for the scattered field. For the modified TDG, we prove [math] and [math]-convergence in the [math] norm for nonabsorbing scatterers. The theoretical results are verified numerically for a discretization based on plane waves, and also investigated numerically for absorbing scatterers (in which case the plane waves are evanescent in the scatterer).
{"title":"Trefftz Discontinuous Galerkin Approximation of an Acoustic Waveguide","authors":"Peter Monk, Manuel Pena, Virginia Selgas","doi":"10.1137/24m1686905","DOIUrl":"https://doi.org/10.1137/24m1686905","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1561-1585, August 2025. <br/> Abstract. We propose a modified Trefftz discontinuous Galerkin (TDG) method for approximating a time-harmonic acoustic scattering problem in an infinitely elongated waveguide. In the waveguide we suppose that there is a bounded, penetrable, and possibly absorbing scatterer. The classical TDG is not applicable when the scatterer is absorbing. Novel features of our modified TDG method are that it is applicable in this case, and it uses a stable treatment of the outgoing radiation condition for the scattered field. For the modified TDG, we prove [math] and [math]-convergence in the [math] norm for nonabsorbing scatterers. The theoretical results are verified numerically for a discretization based on plane waves, and also investigated numerically for absorbing scatterers (in which case the plane waves are evanescent in the scatterer).","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"9 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144669721","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}
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