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The stabilization of linear multistep methods for constrained mechanical systems

APPLIED NUMERICAL MATHEMATICS 1998年第2-4期28卷 143-159页

作者： Amold, M DLR German Aerosp Ctr Inst Robot & Syst Dynam D-82230 Wessling Germany

linear multistep methods from ODE theory may be applied straightforwardly to index-2 DAEs in Hessenberg form if they are strictly stable at infinity (Hairer and Wanner, 1996, Theorem VII.3.6). This condition is very restrictive and excludes, e.g., all higher order Adams methods. In the paper we present an alternative way to apply implicit linear multistep methods to index-2 systems. The convergence of these partitioned linear multistep methods is guaranteed whenever the underlying ODE method is convergent with order p greater than or equal to 3. We discuss the new approach in detail for the application to model equations of constrained mechanical systems. The theoretical results are illustrated by a numerical comparison of multistep methods for index-2 DAEs in Hessenberg form. (C) 1998 Elsevier Science B.V. and IMACS. All rights reserved.

关键词： linear multistep methods higher index DAEs constrained mechanical systems

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DISCOVERY OF DYNAMICS USING linear multistep methods

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2021年第1期59卷 429-455页

作者： Keller, Rachael T. Du, Qiang Columbia Univ Dept Appl Phys & Appl Math New York NY 10027 USA

linear multistep methods (LMMs) are popular time discretization techniques for the numerical solution of differential equations. Traditionally they are applied to solve for the state given the dynamics (the forward problem), but here we consider their application for learning the dynamics given the state (the inverse problem). This repurposing of LMMs is largely motivated by growing interest in data-driven modeling of dynamics, but the behavior and analysis of LMMs for discovery turn out to be significantly different from the well-known, existing theory for the forward problem. Assuming a highly idealized setting of being given the exact state with a zero residual of the discrete dynamics, we establish for the first time a rigorous framework based on refined notions of consistency and stability to yield convergence using LMMs for discovery. When applying these concepts to three popular M-step LMMs, the Adams-Bashforth, Adams-Moulton, and backward differentiation formula schemes, the new theory suggests that Adams-Bashforth for M ranging from 1 and 6, Adams-Moulton for M = 0 and M = 1, and backward differentiation formula for all positive M are convergent, and, otherwise, the methods are not convergent in general. In addition, we provide numerical experiments to both motivate and substantiate our theoretical analysis.

关键词： discovery of dynamics data-driven modeling linear multistep methods stability and convergence root condition learning dynamics artificial intelligence

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ERROR ANALYSIS BASED ON INVERSE MODIFIED DIFFERENTIAL EQUATIONS FOR DISCOVERY OF DYNAMICS USING linear multistep methods AND DEEP LEARNING

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2024年第5期62卷 2087-2120页

作者： Zhu, Aiqing Wu, Sidi Tang, Yifa Chinese Acad Sci Acad Math & Syst Sci Beijing 100190 Peoples R China

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.

关键词： learning dynamics data-driven discovery linear multistep methods deep learning backward error analysis

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Asymptotic stability of linear multistep methods for nonlinear neutral delay differential equations

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APPLIED MATHEMATICS AND COMPUTATION 2009年第1期211卷 95-101页

作者： Hu, Peng Huang, Chengming Wu, Shulin Huazhong Univ Sci & Technol Sch Math & Stat Wuhan 430074 Peoples R China

This paper is concerned with the numerical solution to initial value problems of nonlinear delay differential equations of neutral type. We use A-stable linear multistep methods to compute the numerical solution. The asymptotic stability of the A-stable linear multistep methods when applied to the nonlinear delay differential equations of neutral type is investigated, and it is shown that the A-stable linear multistep methods with linear interpolation are GAS-stable. We validate our conclusions by numerical experiments. (C) 2009 Elsevier Inc. All rights reserved.

关键词： Nonlinear neutral delay differential equations linear multistep methods Asymptotic stability A-stable GAS-stable

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Errors of linear multistep methods and Runge-Kutta methods for singular perturbation problems with delays

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2002年第8-9期44卷 1157-1173页

作者： Gan, SQ Sun, G Zheng, W Tsing Hua Univ Dept Comp Sci & Technol Beijing 100084 Peoples R China Acad Sinica Inst Math Beijing 100080 Peoples R China

This paper is concerned with the error analysis of linear multistep methods and Runge-Kutta methods applied to some classes of one-parameter stiff singularly perturbed problems with delays. We derive the global error estimates of A(alpha)-stable linear multistep methods and algebraically and diagonally stable Runge-Kutta methods with Lagrange interpolation procedure. Numerical experiments confirm our theoretical analysis. (C) 2002 Elsevier Science Ltd. All rights reserved.

关键词： singular perturbation problems delays interpolation linear multistep methods Runge-Kutta methods error analysis

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STABILITY AND CONVERGENCE OF STEPSIZE-DEPENDENT linear multistep methods FOR NONlinear DISSIPATIVE EVOLUTION EQUATIONS IN BANACH SPACE

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Journal of Computational Mathematics 2024年第2期42卷 337-354页

作者： Wansheng Wang Department of Mathematics Shanghai Normal UniversityShanghai 200234China

Stability and global error bounds are studied for a class of stepsize-dependent linear multistep methods for nonlinear evolution equations governed by ω-dissipative vector fields in Banach *** break through the order barrier p≤1 of unconditionally contractive linear multistep methods for dissipative systems,strongly dissipative systems are *** employing the error growth function of the methods,new contractivity and convergence results of stepsize-dependent linear multistep methods on infinite integration intervals are provided for strictly dissipative systems(ω<0)and strongly dissipative *** applications of the main results to several linear multistep methods,including the trapezoidal rule,are *** theoretical results are also illustrated by a set of numerical experiments.

关键词： Nonlinear evolution equation linear multistep methods ω-dissipative operators Stability Convergence Banach space

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On the stability of implicit-explicit linear multistep methods

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APPLIED NUMERICAL MATHEMATICS 1997年第2-3期25卷 193-205页

作者： Frank, J Hundsdorfer, W Verwer, JG CWI P.O. Box 94079 1090 GB Amsterdam The Netherlands

In many applications, such as atmospheric chemistry, large systems of ordinary differential equations (ODEs) with both stiff and nonstiff parts have to be solved numerically. A popular approach in such cases is to integrate the stiff parts implicitly and the nonstiff parts explicitly. In this paper we study a class of implicit-explicit (IMEX) linear multistep methods intended for such applications. The paper focuses on the linear stability of popular second order methods like extrapolated BDF, Crank-Nicolson leap-frog and a particular class of Adams methods. We present results for problems with decoupled eigenvalues and comment on some specific CFL restrictions associated with advection terms. (C) 1997 Elsevier Science B.V.

关键词： implicit-explicit methods linear multistep methods method of lines stability

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STRONG STABILITY PRESERVING EXPLICIT linear multistep methods WITH VARIABLE STEP SIZE

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2016年第5期54卷 2799-2832页

作者： Hadjimichael, Yiannis Ketcheson, David I. Loczi, Lajos Nemeth, Adrian KAUST Comp Elect & Math Sci & Engn Div Jeddah 239556900 Saudi Arabia Szechenyi Istvan Univ H-9026 Gyor Hungary

Strong stability preserving (SSP) methods are designed primarily for time integration of nonlinear hyperbolic PDEs, for which the permissible SSP step size varies from one step to the next. We develop the first SSP linear multistep methods (of order two and three) with variable step size, and prove their optimality, stability, and convergence. The choice of step size for multistep SSP methods is an interesting problem because the allowable step size depends on the SSP coefficient, which in turn depends on the chosen step sizes. The description of the methods includes an optimal step-size strategy. We prove sharp upper bounds on the allowable step size for explicit SSP linear multistep methods and show the existence of methods with arbitrarily high order of accuracy. The effectiveness of the methods is demonstrated through numerical examples.

关键词： strong stability preservation monotonicity linear multistep methods variable step size time integration

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Data-driven sparse identification of nonlinear dynamical systems using linear multistep methods

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CALCOLO 2023年第1期60卷 1-26页

作者： Chen, Hao Chongqing Normal Univ Coll Math Sci Chongqing Peoples R China

linear multistep methods (LMMs) are popular time discretization schemes for solving the forward problem on differential equations. Recently, LMMs together with deep neural networks have been shown to successfully discover dynamical systems from data. In this work, we propose a class of LMM-based sparse regression approaches for the discovery of nonlinear dynamical systems. The work builds on the sparse identification of nonlinear dynamics (SINDy) framework presented in Brunton et al. (Proc Natl Acad Sci USA 113: 3932-3937, 2016), allowing closed form expression for the governing equations and therefore the resulting data-driven model can give insights into the underlying physics. Compared to the standard SINDy algorithm, the proposed LMM-based SINDy approach allows for more accurate and robust model recovery from data with a wide range of noise levels, without requiring pointwise derivative approximations and conventional noise filtering. Numerical results are presented to demonstrate the effectiveness of the proposed method.

关键词： Data-driven discovery Nonlinear dynamics Sparse regression Generalized least squares linear multistep methods

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DELAY-DEPENDENT STABILITY OF linear multistep methods FOR NEUTRAL SYSTEMS WITH DISTRIBUTED DELAYS

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Journal of Computational Mathematics 2022年第3期40卷 484-498页

作者： Yuhao Cong Shouyan Wu Shanghai Customs College Shanghai 201204China Department of Mathematics Zhejiang International Studies UniversityHangzhou 310023China Department of Mathematics Shanghai Normal UniversityShanghai 200234China

This paper considers the asymptotic stability of linear multistep(LM)methods for neutral systems with distributed *** particular,several sufficient conditions for delay-dependent stability of numerical solutions are obtained based on the argument *** quadrature formulae are used to compute the *** algorithm is proposed to examine the delay-dependent stability of numerical *** numerical examples are performed to verify the theoretical results.

关键词： Neutral systems with distributed delays linear multistep methods Delaydependent stability Argument principle

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