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}
A. Brunk, J. Giesselmann, M. Lukáčová-Medvi[math]ová
SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1540-1560, August 2025. Abstract. In this work, we derive a [math]-robust a posteriori error estimator for finite element approximations of the Allen–Cahn equation with variable nondegenerate mobility. The estimator utilizes spectral estimates for the linearized steady part of the differential operator as well as a conditional stability estimate based on a weighted sum of Bregman distances, based on the energy and a functional related to the mobility. A suitable reconstruction of the numerical solution in the stability estimate leads to a fully computable estimator.
{"title":"A Posteriori Error Control for the Allen–Cahn Equation with Variable Mobility","authors":"A. Brunk, J. Giesselmann, M. Lukáčová-Medvi[math]ová","doi":"10.1137/24m1646406","DOIUrl":"https://doi.org/10.1137/24m1646406","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1540-1560, August 2025. <br/> Abstract. In this work, we derive a [math]-robust a posteriori error estimator for finite element approximations of the Allen–Cahn equation with variable nondegenerate mobility. The estimator utilizes spectral estimates for the linearized steady part of the differential operator as well as a conditional stability estimate based on a weighted sum of Bregman distances, based on the energy and a functional related to the mobility. A suitable reconstruction of the numerical solution in the stability estimate leads to a fully computable estimator.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"5 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144645433","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}
Bangti Jin, Qimeng Quan, Barbara Wohlmuth, Zhi Zhou
SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1512-1539, August 2025. Abstract. In this work, we investigate a quasilinear subdiffusion model which involves a fractional derivative of order [math] in time and a nonlinear diffusion coefficient. First, using smoothing properties of solution operators for linear subdiffusion and a perturbation argument, we prove several new pointwise-in-time Sobolev regularity estimates that are useful for numerical analysis. Then we develop a time-stepping scheme to solve quasilinear subdiffusion, based on convolution quadrature generated by the second-order backward differentiation formula with a correction at the first step. Further, we establish that the convergence order of the scheme is [math] without imposing any additional assumption on the regularity of the solution, which is high-order in the sense that its convergence rate is higher than the first-order convergence of the vanilla scheme. The analysis relies on refined Sobolev regularity of the nonlinear perturbation remainder and smoothing properties of discrete solution operators. Several numerical experiments in two space dimensions are presented to show the sharpness of the error estimate.
{"title":"Regularity Analysis and High-Order Time Stepping Scheme for Quasilinear Subdiffusion","authors":"Bangti Jin, Qimeng Quan, Barbara Wohlmuth, Zhi Zhou","doi":"10.1137/23m159531x","DOIUrl":"https://doi.org/10.1137/23m159531x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1512-1539, August 2025. <br/> Abstract. In this work, we investigate a quasilinear subdiffusion model which involves a fractional derivative of order [math] in time and a nonlinear diffusion coefficient. First, using smoothing properties of solution operators for linear subdiffusion and a perturbation argument, we prove several new pointwise-in-time Sobolev regularity estimates that are useful for numerical analysis. Then we develop a time-stepping scheme to solve quasilinear subdiffusion, based on convolution quadrature generated by the second-order backward differentiation formula with a correction at the first step. Further, we establish that the convergence order of the scheme is [math] without imposing any additional assumption on the regularity of the solution, which is high-order in the sense that its convergence rate is higher than the first-order convergence of the vanilla scheme. The analysis relies on refined Sobolev regularity of the nonlinear perturbation remainder and smoothing properties of discrete solution operators. Several numerical experiments in two space dimensions are presented to show the sharpness of the error estimate.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"84 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144645495","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 1482-1511, August 2025. Abstract. We construct a convolution quadrature (CQ) scheme for the quasilinear subdiffusion equation of order [math] and supply it with the fast and oblivious implementation. In particular, we find a condition for the CQ to be admissible and discretize the spatial part of the equation with the finite element method. We prove the unconditional stability and convergence of the scheme and find a bound on the error. Our estimates are globally optimal for all [math] and pointwise for [math] in the sense that they reduce to the well-known results for the linear equation. For the semilinear case, our estimates are optimal both globally and locally. As a passing result, we also obtain a discrete Grönwall inequality for the CQ, which is a crucial ingredient in our convergence proof based on the energy method. The paper concludes with numerical examples verifying convergence and computation time reduction when using fast and oblivious quadrature.
SIAM Journal on Numerical Analysis, vol . 63, Issue 4, Page 1482-1511, August 2025。摘要。本文构造了一类准线性次扩散方程的卷积正交（CQ）格式，并提供了快速且遗忘的实现。特别地，我们找到了CQ允许的一个条件，并用有限元方法将方程的空间部分离散化。证明了该方案的无条件稳定性和收敛性，并找到了误差的一个界。我们的估计对所有[数学]和[数学]来说都是全局最优的，在某种意义上，它们减少到众所周知的线性方程的结果。对于半线性的情况，我们的估计在全局和局部都是最优的。作为一个合格的结果，我们还得到了CQ的离散Grönwall不等式，这是我们基于能量法的收敛性证明的关键因素。最后用数值算例验证了快速无关正交的收敛性和减少了计算时间。
{"title":"Convolution Quadrature for the Quasilinear Subdiffusion Equation","authors":"Maria López-Fernández, Łukasz Płociniczak","doi":"10.1137/23m161450x","DOIUrl":"https://doi.org/10.1137/23m161450x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1482-1511, August 2025. <br/> Abstract. We construct a convolution quadrature (CQ) scheme for the quasilinear subdiffusion equation of order [math] and supply it with the fast and oblivious implementation. In particular, we find a condition for the CQ to be admissible and discretize the spatial part of the equation with the finite element method. We prove the unconditional stability and convergence of the scheme and find a bound on the error. Our estimates are globally optimal for all [math] and pointwise for [math] in the sense that they reduce to the well-known results for the linear equation. For the semilinear case, our estimates are optimal both globally and locally. As a passing result, we also obtain a discrete Grönwall inequality for the CQ, which is a crucial ingredient in our convergence proof based on the energy method. The paper concludes with numerical examples verifying convergence and computation time reduction when using fast and oblivious quadrature.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"9 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144630011","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 1454-1481, August 2025. Abstract. A new approach is developed to study the convergence of parametric finite element approximations to the mean curvature flow of closed surfaces in three-dimensional space. In this approach, the error analysis is conducted by comparing the numerical solution to a dynamic Ritz projection of the mean curvature flow introduced in this paper rather than an interpolation of the mean curvature flow, as commonly used in the literature. The errors associated with the dynamic Ritz projection in approximating the mean curvature flow are established in the [math] and [math] norms. Leveraging these results, optimal-order convergence of parametric finite element methods for mean curvature flow of closed surfaces in the [math] norm is proved, including the convergence of parametric finite element methods with piecewise linear finite elements.
{"title":"Dynamic Ritz Projection of Mean Curvature Flow and Optimal [math] Convergence of Parametric FEM","authors":"Buyang Li, Rong Tang","doi":"10.1137/24m1689053","DOIUrl":"https://doi.org/10.1137/24m1689053","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1454-1481, August 2025. <br/> Abstract. A new approach is developed to study the convergence of parametric finite element approximations to the mean curvature flow of closed surfaces in three-dimensional space. In this approach, the error analysis is conducted by comparing the numerical solution to a dynamic Ritz projection of the mean curvature flow introduced in this paper rather than an interpolation of the mean curvature flow, as commonly used in the literature. The errors associated with the dynamic Ritz projection in approximating the mean curvature flow are established in the [math] and [math] norms. Leveraging these results, optimal-order convergence of parametric finite element methods for mean curvature flow of closed surfaces in the [math] norm is proved, including the convergence of parametric finite element methods with piecewise linear finite elements.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"101 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144629958","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}
Charles L. Epstein, Fredrik Fryklund, Shidong Jiang
SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1427-1453, August 2025. Abstract. A new scheme is proposed to construct an [math]-times differentiable function extension of an [math]-times differentiable function defined on a smooth domain, [math] in [math]-dimensions. The extension scheme relies on an explicit formula consisting of a linear combination of [math] function values in [math] which extends the function along directions normal to the boundary. Smoothness tangent to the boundary is automatic. The performance of the scheme is illustrated by using function extension as part of a numerical solver for the Poisson equation on domains with complex geometry in both two and three dimensions. Although the cost of extending the function increases mildly with the extension order, it remains a small fraction of the overall algorithm. Moreover, the modest additional work required for high order function extensions leads to considerably more accurate solutions of the partial differential equation.
{"title":"An Accurate and Efficient Scheme for Function Extension on Smooth Domains","authors":"Charles L. Epstein, Fredrik Fryklund, Shidong Jiang","doi":"10.1137/23m1622064","DOIUrl":"https://doi.org/10.1137/23m1622064","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1427-1453, August 2025. <br/> Abstract. A new scheme is proposed to construct an [math]-times differentiable function extension of an [math]-times differentiable function defined on a smooth domain, [math] in [math]-dimensions. The extension scheme relies on an explicit formula consisting of a linear combination of [math] function values in [math] which extends the function along directions normal to the boundary. Smoothness tangent to the boundary is automatic. The performance of the scheme is illustrated by using function extension as part of a numerical solver for the Poisson equation on domains with complex geometry in both two and three dimensions. Although the cost of extending the function increases mildly with the extension order, it remains a small fraction of the overall algorithm. Moreover, the modest additional work required for high order function extensions leads to considerably more accurate solutions of the partial differential equation.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"35 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144603408","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}
Abhijit Biswas, David I. Ketcheson, Steven Roberts, Benjamin Seibold, David Shirokoff
SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1398-1426, August 2025. Abstract. Explicit Runge–Kutta (RK) methods are susceptible to a reduction in the observed order of convergence when applied to an initial boundary value problem with time-dependent boundary conditions. We study conditions on explicit RK methods that guarantee high order convergence for linear problems; we refer to these conditions as weak stage order conditions. We prove a general relationship between the method’s order, weak stage order, and number of stages. We derive explicit RK methods with high weak stage order and demonstrate, through numerical tests, that they avoid the order reduction phenomenon up to any order for linear problems and up to order three for nonlinear problems.
{"title":"Explicit Runge–Kutta Methods that Alleviate Order Reduction","authors":"Abhijit Biswas, David I. Ketcheson, Steven Roberts, Benjamin Seibold, David Shirokoff","doi":"10.1137/23m1606812","DOIUrl":"https://doi.org/10.1137/23m1606812","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1398-1426, August 2025. <br/> Abstract. Explicit Runge–Kutta (RK) methods are susceptible to a reduction in the observed order of convergence when applied to an initial boundary value problem with time-dependent boundary conditions. We study conditions on explicit RK methods that guarantee high order convergence for linear problems; we refer to these conditions as weak stage order conditions. We prove a general relationship between the method’s order, weak stage order, and number of stages. We derive explicit RK methods with high weak stage order and demonstrate, through numerical tests, that they avoid the order reduction phenomenon up to any order for linear problems and up to order three for nonlinear problems.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"21 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144586772","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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