Modified Patankar schemes are linearly implicit time integration methods designed to be unconditionally positive and conservative. In the present work we extend the Patankar-type approach to linear multistep methods a...
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Modified Patankar schemes are linearly implicit time integration methods designed to be unconditionally positive and conservative. In the present work we extend the Patankar-type approach to linear multistep methods and prove that the resulting discretizations retain, with no restrictions on the step size, the positivity of the solution and the linear invariant of the continuous-time system. Moreover, we provide results on arbitrarily high order of convergence and we introduce an embedding technique for the Patankar weights denominators to achieve it.
The numerical approximation of solutions of differential equations has been and continues to be one of the principal concerns of numerical analysis. linear multistep methods and, in particular, backward differentiatio...
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The numerical approximation of solutions of differential equations has been and continues to be one of the principal concerns of numerical analysis. linear multistep methods and, in particular, backward differentiation formulae (BDFs) are frequently used for the numerical integration of stiff initial value problems. Such stiff problems appear in a variety of applications. While the intuitive meaning of stiffness is clear to all specialists, there has been much controversy about its correct mathematical definition. We present a historical development of the concept of stiffness. A survey of convergence results for special classes of stiff problems based on these different concepts of stiffness is given, e.g., for linear, stiff systems, problems in singular perturbation form, nonautonomous stiff systems, and rather general nonlinear stiff problems. Different approaches proving convergence of linear multistep methods applied to stiff initial value problems are introduced. It is further indicated that the corresponding proofs for singular perturbation problems are compatible with a nonlinear transformation and thus convergence of a quite general class of nonlinear problems seems to be covered. (C) 2005 Elsevier Ltd. All rights reserved.
We are interested in high-order linearmultistep schemes for time discretization of adjoint equations arising within optimal control problems. First we consider optimal control problems for ordinary differential equat...
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We are interested in high-order linearmultistep schemes for time discretization of adjoint equations arising within optimal control problems. First we consider optimal control problems for ordinary differential equations and show loss of accuracy for Adams-Moulton and Adams-Bashforth methods, whereas BDF methods preserve high-order accuracy. Subsequently we extend these results to semi-Lagrangian discretizations of hyperbolic relaxation systems. Computational results illustrate theoretical findings. (C) 2019 Elsevier Inc. All rights reserved.
This paper deals with the convergence and stability of linear multistep methods for a class of linear impulsive delay differential equations. Numerical experiments show that the Simpson's Rule and two-step BDF met...
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This paper deals with the convergence and stability of linear multistep methods for a class of linear impulsive delay differential equations. Numerical experiments show that the Simpson's Rule and two-step BDF method are of order p = 0 when applied to impulsive delay differential equations. An improved linearmultistep numerical process is proposed. Convergence and stability conditions of the numerical solutions are given in the paper. Numerical experiments are given in the end to illustrate the conclusion. (C) 2017 Elsevier Inc. All rights reserved.
linear multistep methods (LMMs) are written as irreducible general linearmethods (GLMs). A-stable LMMs are shown to be algebraically stable GLMs for strictly positive definite G-matrices. Optimal order error bounds, ...
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linear multistep methods (LMMs) are written as irreducible general linearmethods (GLMs). A-stable LMMs are shown to be algebraically stable GLMs for strictly positive definite G-matrices. Optimal order error bounds, independent of stiffness, are derived for A-stable methods, without considering one-leg methods (OLMs). As a GLM, the OLM is shown to be the transpose of the LMM. For A-stable methods, the LMM G-matrix is the inverse of the OLM G-matrix. Examples of G-symplectic LMMs are given.
Abstract: A general class of linear multistep methods is presented for numerically solving first- and second-kind Volterra integral equations, and Volterra integro-differential equations. These so-called VLM m...
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Abstract: A general class of linear multistep methods is presented for numerically solving first- and second-kind Volterra integral equations, and Volterra integro-differential equations. These so-called VLMmethods, which include the well-known direct quadrature methods, allow for a unified treatment of the problems of consistency and convergence, and have an analogue in linear multistep methods for ODEs, as treated in any textbook on computational methods in ordinary differential equations. General consistency and convergence results are presented (and proved in an Appendix), together with results of numerical experiments which support the theory.
The paper reviews results on rigorous proofs for stability properties of classes of linear multistep methods (LMMs) used either as IVMs or as BVMs. The considered classes are not only the well-known classical ones (BD...
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The paper reviews results on rigorous proofs for stability properties of classes of linear multistep methods (LMMs) used either as IVMs or as BVMs. The considered classes are not only the well-known classical ones (BDF, Adams, ...) along with their BVM correspondent, but also those which were considered unstable as IVMs, but stable as BVMs. Among the latter we find two classes which deserve attention because of their peculiarity: the TOMs (top order methods) which have the highest order allowed to a LMM and the Bs-LMMs which have the property to carry with each method its natural continuous extension. (C) 2006 Elsevier B.V. All rights reserved.
In this paper, starting from a sequence of results which can be traced back to I. J. Schoenberg, we analyze a class of spline collocation methods for the numerical solution of ordinary differential equations (ODEs) wi...
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In this paper, starting from a sequence of results which can be traced back to I. J. Schoenberg, we analyze a class of spline collocation methods for the numerical solution of ordinary differential equations (ODEs) with collocation points coinciding with the knots. Such collocation methods are naturally associated to a special class of linear multistep methods, here called B-spline (BS) methods, which are able to generate the spline values at the knots. We prove that, provided the additional conditions are appropriately chosen, such methods are all convergent and A-stable. The convergence property of the BS methods is naturally inherited by the related spline extensions, which, by the way, are easily and safely computable using their B-spline representation.
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 r...
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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 multistepmethods for index-2 DAEs in Hessenberg form. (C) 1998 Elsevier Science B.V. and IMACS. All rights reserved.
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 pr...
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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.
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