SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2172-2195, October 2024. Abstract. A novel evolving surface finite element method, based on a novel equivalent formulation of the continuous problem, is proposed for computing the evolution of a closed hypersurface moving under a prescribed velocity field in two- and three-dimensional spaces. The method improves the mesh quality of the approximate surface by minimizing the rate of deformation using an artificial tangential motion. The transport evolution equations of the normal vector and the extrinsic Weingarten matrix are derived and coupled with the surface evolution equations to ensure stability and convergence of the numerical approximations. Optimal-order convergence of the semidiscrete evolving surface finite element method is proved for finite elements of degree [math]. Numerical examples are provided to illustrate the convergence of the proposed method and its effectiveness in improving mesh quality on the approximate evolving surface.
{"title":"A Convergent Evolving Finite Element Method with Artificial Tangential Motion for Surface Evolution under a Prescribed Velocity Field","authors":"Genming Bai, Jiashun Hu, Buyang Li","doi":"10.1137/23m156968x","DOIUrl":"https://doi.org/10.1137/23m156968x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2172-2195, October 2024. <br/> Abstract. A novel evolving surface finite element method, based on a novel equivalent formulation of the continuous problem, is proposed for computing the evolution of a closed hypersurface moving under a prescribed velocity field in two- and three-dimensional spaces. The method improves the mesh quality of the approximate surface by minimizing the rate of deformation using an artificial tangential motion. The transport evolution equations of the normal vector and the extrinsic Weingarten matrix are derived and coupled with the surface evolution equations to ensure stability and convergence of the numerical approximations. Optimal-order convergence of the semidiscrete evolving surface finite element method is proved for finite elements of degree [math]. Numerical examples are provided to illustrate the convergence of the proposed method and its effectiveness in improving mesh quality on the approximate evolving surface.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"329 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142235036","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 62, Issue 5, Page 2143-2171, October 2024. Abstract. The coupling of nonconservative hyperbolic systems at a static interface has been a delicate issue as common approaches rely on the Lax-curves of the systems, which are not always available. To address this a new linear relaxation system is introduced, in which a nonlocal source term accounts for the nonconservative product of the original system. Using an asymptotic analysis the relaxation limit and its stability are investigated in the uncoupled setting. It is shown that the path-conservative Lax–Friedrichs scheme arises from a discrete limit of an implicit-explicit scheme for the relaxation system. Employing the relaxation approach, a novel technique to couple two nonconservative systems under a large class of coupling conditions is established. A particular coupling strategy motivated from conservative Kirchhoff conditions is introduced and a corresponding Riemann solver provided. A fully discrete scheme for coupled nonconservative products is derived and studied in terms of path conservation. Numerical experiments applying the approach to a coupled model of vascular blood flow are presented.
{"title":"Numerical Schemes for Coupled Systems of Nonconservative Hyperbolic Equations","authors":"Niklas Kolbe, Michael Herty, Siegfried Müller","doi":"10.1137/23m1615176","DOIUrl":"https://doi.org/10.1137/23m1615176","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2143-2171, October 2024. <br/> Abstract. The coupling of nonconservative hyperbolic systems at a static interface has been a delicate issue as common approaches rely on the Lax-curves of the systems, which are not always available. To address this a new linear relaxation system is introduced, in which a nonlocal source term accounts for the nonconservative product of the original system. Using an asymptotic analysis the relaxation limit and its stability are investigated in the uncoupled setting. It is shown that the path-conservative Lax–Friedrichs scheme arises from a discrete limit of an implicit-explicit scheme for the relaxation system. Employing the relaxation approach, a novel technique to couple two nonconservative systems under a large class of coupling conditions is established. A particular coupling strategy motivated from conservative Kirchhoff conditions is introduced and a corresponding Riemann solver provided. A fully discrete scheme for coupled nonconservative products is derived and studied in terms of path conservation. Numerical experiments applying the approach to a coupled model of vascular blood flow are presented.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"130 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142166257","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 62, Issue 5, Page 2121-2142, October 2024. Abstract. This paper studies the discretization of a homogenization and dimension reduction model for the elastic deformation of microstructured thin plates proposed by Hornung, Neukamm, and Velčić [Calc. Var. Partial Differential Equations, 51 (2014), pp. 677–699]. Thereby, a nonlinear bending energy is based on a homogenized quadratic form which acts on the second fundamental form associated with the elastic deformation. Convergence is proved for a multi-affine finite element discretization of the involved three-dimensional microscopic cell problems and a discrete Kirchhoff triangle discretization of the two-dimensional isometry-constrained macroscopic problem. Finally, the convergence properties are numerically verified in selected test cases and qualitatively compared with deformation experiments for microstructured sheets of paper.
{"title":"Two-Scale Finite Element Approximation of a Homogenized Plate Model","authors":"Martin Rumpf, Stefan Simon, Christoph Smoch","doi":"10.1137/23m1596272","DOIUrl":"https://doi.org/10.1137/23m1596272","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2121-2142, October 2024. <br/> Abstract. This paper studies the discretization of a homogenization and dimension reduction model for the elastic deformation of microstructured thin plates proposed by Hornung, Neukamm, and Velčić [Calc. Var. Partial Differential Equations, 51 (2014), pp. 677–699]. Thereby, a nonlinear bending energy is based on a homogenized quadratic form which acts on the second fundamental form associated with the elastic deformation. Convergence is proved for a multi-affine finite element discretization of the involved three-dimensional microscopic cell problems and a discrete Kirchhoff triangle discretization of the two-dimensional isometry-constrained macroscopic problem. Finally, the convergence properties are numerically verified in selected test cases and qualitatively compared with deformation experiments for microstructured sheets of paper.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"9 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142166291","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 62, Issue 5, Page 2087-2120, October 2024. Abstract. Along with the practical success of the discovery of dynamics using deep learning, the theoretical analysis of this approach has attracted increasing attention. Prior works have established the grid error estimation with auxiliary conditions for the discovery of dynamics using linear multistep methods and deep learning. And we extend the existing error analysis in this work. We first introduce the concept of inverse modified differential equations (IMDE) for linear multistep methods and show that the learned model returns a close approximation of the IMDE. Based on the IMDE, we prove that the error between the discovered system and the target system is bounded by the sum of the LMM discretization error and the learning loss. Furthermore, the learning loss is quantified by combining the approximation and generalization theories of neural networks, and thereby we obtain the priori error estimates. Several numerical experiments are performed to verify the theoretical analysis.
{"title":"Error Analysis Based on Inverse Modified Differential Equations for Discovery of Dynamics Using Linear Multistep Methods and Deep Learning","authors":"Aiqing Zhu, Sidi Wu, Yifa Tang","doi":"10.1137/22m152373x","DOIUrl":"https://doi.org/10.1137/22m152373x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2087-2120, October 2024. <br/> Abstract. Along with the practical success of the discovery of dynamics using deep learning, the theoretical analysis of this approach has attracted increasing attention. Prior works have established the grid error estimation with auxiliary conditions for the discovery of dynamics using linear multistep methods and deep learning. And we extend the existing error analysis in this work. We first introduce the concept of inverse modified differential equations (IMDE) for linear multistep methods and show that the learned model returns a close approximation of the IMDE. Based on the IMDE, we prove that the error between the discovered system and the target system is bounded by the sum of the LMM discretization error and the learning loss. Furthermore, the learning loss is quantified by combining the approximation and generalization theories of neural networks, and thereby we obtain the priori error estimates. Several numerical experiments are performed to verify the theoretical analysis.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"17 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142138386","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}
Lun Ji, Alexander Ostermann, Frédéric Rousset, Katharina Schratz
SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2071-2086, October 2024. Abstract. For the numerical solution of the cubic nonlinear Schrödinger equation with periodic boundary conditions, a pseudospectral method in space combined with a filtered Lie splitting scheme in time is considered. This scheme is shown to converge even for initial data with very low regularity. In particular, for data in [math], where [math], convergence of order [math] is proved in [math]. Here [math] denotes the time step size and [math] the number of Fourier modes considered. The proof of this result is carried out in an abstract framework of discrete Bourgain spaces; the final convergence result, however, is given in [math]. The stated convergence behavior is illustrated by several numerical examples.
{"title":"Low Regularity Full Error Estimates for the Cubic Nonlinear Schrödinger Equation","authors":"Lun Ji, Alexander Ostermann, Frédéric Rousset, Katharina Schratz","doi":"10.1137/23m1619617","DOIUrl":"https://doi.org/10.1137/23m1619617","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2071-2086, October 2024. <br/> Abstract. For the numerical solution of the cubic nonlinear Schrödinger equation with periodic boundary conditions, a pseudospectral method in space combined with a filtered Lie splitting scheme in time is considered. This scheme is shown to converge even for initial data with very low regularity. In particular, for data in [math], where [math], convergence of order [math] is proved in [math]. Here [math] denotes the time step size and [math] the number of Fourier modes considered. The proof of this result is carried out in an abstract framework of discrete Bourgain spaces; the final convergence result, however, is given in [math]. The stated convergence behavior is illustrated by several numerical examples.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"14 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142130829","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 62, Issue 4, Page 2048-2070, August 2024. Abstract. We present new Dirichlet–Neumann and Neumann–Dirichlet algorithms with a time domain decomposition applied to unconstrained parabolic optimal control problems. After a spatial semidiscretization, we use the Lagrange multiplier approach to derive a coupled forward-backward optimality system, which can then be solved using a time domain decomposition. Due to the forward-backward structure of the optimality system, three variants can be found for the Dirichlet–Neumann and Neumann–Dirichlet algorithms. We analyze their convergence behavior and determine the optimal relaxation parameter for each algorithm. Our analysis reveals that the most natural algorithms are actually only good smoothers, and there are better choices which lead to efficient solvers. We illustrate our analysis with numerical experiments.
{"title":"New Time Domain Decomposition Methods for Parabolic Optimal Control Problems I: Dirichlet–Neumann and Neumann–Dirichlet Algorithms","authors":"Martin J. Gander, Liu-Di Lu","doi":"10.1137/23m1584502","DOIUrl":"https://doi.org/10.1137/23m1584502","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 2048-2070, August 2024. <br/> Abstract. We present new Dirichlet–Neumann and Neumann–Dirichlet algorithms with a time domain decomposition applied to unconstrained parabolic optimal control problems. After a spatial semidiscretization, we use the Lagrange multiplier approach to derive a coupled forward-backward optimality system, which can then be solved using a time domain decomposition. Due to the forward-backward structure of the optimality system, three variants can be found for the Dirichlet–Neumann and Neumann–Dirichlet algorithms. We analyze their convergence behavior and determine the optimal relaxation parameter for each algorithm. Our analysis reveals that the most natural algorithms are actually only good smoothers, and there are better choices which lead to efficient solvers. We illustrate our analysis with numerical experiments.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"4 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142042382","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 62, Issue 4, Page 2025-2047, August 2024. Abstract. Statistical inverse learning aims at recovering an unknown function [math] from randomly scattered and possibly noisy point evaluations of another function [math], connected to [math] via an ill-posed mathematical model. In this paper we blend statistical inverse learning theory with the classical regularization strategy of applying finite-dimensional projections. Our key finding is that coupling the number of random point evaluations with the choice of projection dimension, one can derive probabilistic convergence rates for the reconstruction error of the maximum likelihood (ML) estimator. Convergence rates in expectation are derived with a ML estimator complemented with a norm-based cutoff operation. Moreover, we prove that the obtained rates are minimax optimal.
{"title":"Least Squares Approximations in Linear Statistical Inverse Learning Problems","authors":"Tapio Helin","doi":"10.1137/22m1538600","DOIUrl":"https://doi.org/10.1137/22m1538600","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 2025-2047, August 2024. <br/> Abstract. Statistical inverse learning aims at recovering an unknown function [math] from randomly scattered and possibly noisy point evaluations of another function [math], connected to [math] via an ill-posed mathematical model. In this paper we blend statistical inverse learning theory with the classical regularization strategy of applying finite-dimensional projections. Our key finding is that coupling the number of random point evaluations with the choice of projection dimension, one can derive probabilistic convergence rates for the reconstruction error of the maximum likelihood (ML) estimator. Convergence rates in expectation are derived with a ML estimator complemented with a norm-based cutoff operation. Moreover, we prove that the obtained rates are minimax optimal.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"32 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142042381","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 62, Issue 4, Page 2004-2024, August 2024. Abstract. We propose and analyze a novel approach to construct structure preserving approximations for the Poisson–Nernst–Planck equations, focusing on the positivity preserving and mass conservation properties. The strategy consists of a standard time marching step with a projection (or correction) step to satisfy the desired physical constraints (positivity and mass conservation). Based on the [math] projection, we construct a second order Crank–Nicolson type finite difference scheme, which is linear (exclude the very efficient [math] projection part), positivity preserving, and mass conserving. Rigorous error estimates in the [math] norm are established, which are both second order accurate in space and time. The other choice of projection, e.g., [math] projection, is discussed. Numerical examples are presented to verify the theoretical results and demonstrate the efficiency of the proposed method.
{"title":"Positivity Preserving and Mass Conservative Projection Method for the Poisson–Nernst–Planck Equation","authors":"Fenghua Tong, Yongyong Cai","doi":"10.1137/23m1581649","DOIUrl":"https://doi.org/10.1137/23m1581649","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 2004-2024, August 2024. <br/> Abstract. We propose and analyze a novel approach to construct structure preserving approximations for the Poisson–Nernst–Planck equations, focusing on the positivity preserving and mass conservation properties. The strategy consists of a standard time marching step with a projection (or correction) step to satisfy the desired physical constraints (positivity and mass conservation). Based on the [math] projection, we construct a second order Crank–Nicolson type finite difference scheme, which is linear (exclude the very efficient [math] projection part), positivity preserving, and mass conserving. Rigorous error estimates in the [math] norm are established, which are both second order accurate in space and time. The other choice of projection, e.g., [math] projection, is discussed. Numerical examples are presented to verify the theoretical results and demonstrate the efficiency of the proposed method.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"29 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142007533","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}
Yassine Boubendir, Jake Brusca, Brittany F. Hamfeldt, Tadanaga Takahashi
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1979-2003, August 2024. Abstract. We introduce a new overlapping domain decomposition method (DDM) to solve fully nonlinear elliptic partial differential equations (PDEs) approximated with monotone schemes. While DDMs have been extensively studied for linear problems, their application to fully nonlinear PDEs remains limited in the literature. To address this gap, we establish a proof of global convergence of these new iterative algorithms using a discrete comparison principle argument. We also provide a specific implementation for the Monge–Ampère equation. Several numerical tests are performed to validate the convergence theorem. These numerical experiments involve examples of varying regularity. Computational experiments show that method is efficient, robust, and requires relatively few iterations to converge. The results reveal great potential for DDM methods to lead to highly efficient and parallelizable solvers for large-scale problems that are computationally intractable using existing solution methods.
{"title":"Domain Decomposition Methods for the Monge–Ampère Equation","authors":"Yassine Boubendir, Jake Brusca, Brittany F. Hamfeldt, Tadanaga Takahashi","doi":"10.1137/23m1576839","DOIUrl":"https://doi.org/10.1137/23m1576839","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1979-2003, August 2024. <br/> Abstract. We introduce a new overlapping domain decomposition method (DDM) to solve fully nonlinear elliptic partial differential equations (PDEs) approximated with monotone schemes. While DDMs have been extensively studied for linear problems, their application to fully nonlinear PDEs remains limited in the literature. To address this gap, we establish a proof of global convergence of these new iterative algorithms using a discrete comparison principle argument. We also provide a specific implementation for the Monge–Ampère equation. Several numerical tests are performed to validate the convergence theorem. These numerical experiments involve examples of varying regularity. Computational experiments show that method is efficient, robust, and requires relatively few iterations to converge. The results reveal great potential for DDM methods to lead to highly efficient and parallelizable solvers for large-scale problems that are computationally intractable using existing solution methods.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"14 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141980985","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 62, Issue 4, Page 1956-1978, August 2024. Abstract. In this article, we employ the construction of the time-marching discontinuous Petrov–Galerkin (DPG) scheme we developed for linear problems to derive high-order multistage DPG methods for nonlinear systems of ordinary differential equations. The methodology extends to abstract evolution equations in Banach spaces, including a class of nonlinear partial differential equations. We present three nested multistage methods: the hybrid Euler method and the two- and three-stage DPG methods. We employ a linearization of the problem as in exponential Rosenbrock methods, so we need to compute exponential actions of the Jacobian that change from time step to time step. The key point of our construction is that one of the stages can be postprocessed from another without an extra exponential step. Therefore, the class of methods we introduce is computationally cheaper than the classical exponential Rosenbrock methods. We provide a full convergence proof to show that the methods are second-, third-, and fourth-order accurate, respectively. We test the convergence in time of our methods on a 2D+time semilinear partial differential equation after a semidiscretization in space.
{"title":"Multistage Discontinuous Petrov–Galerkin Time-Marching Scheme for Nonlinear Problems","authors":"Judit Muñoz-Matute, Leszek Demkowicz","doi":"10.1137/23m1598088","DOIUrl":"https://doi.org/10.1137/23m1598088","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1956-1978, August 2024. <br/> Abstract. In this article, we employ the construction of the time-marching discontinuous Petrov–Galerkin (DPG) scheme we developed for linear problems to derive high-order multistage DPG methods for nonlinear systems of ordinary differential equations. The methodology extends to abstract evolution equations in Banach spaces, including a class of nonlinear partial differential equations. We present three nested multistage methods: the hybrid Euler method and the two- and three-stage DPG methods. We employ a linearization of the problem as in exponential Rosenbrock methods, so we need to compute exponential actions of the Jacobian that change from time step to time step. The key point of our construction is that one of the stages can be postprocessed from another without an extra exponential step. Therefore, the class of methods we introduce is computationally cheaper than the classical exponential Rosenbrock methods. We provide a full convergence proof to show that the methods are second-, third-, and fourth-order accurate, respectively. We test the convergence in time of our methods on a 2D+time semilinear partial differential equation after a semidiscretization in space.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"13 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141910473","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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