On approximate implicit Taylor methods for ordinary differential equations
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On approximate implicit Taylor methods for ordinary differential equations Antonio Baeza1 · Raimund Bürger2 · María del Carmen Martí1 · Pep Mulet1 · David Zorío3 Received: 27 July 2019 / Revised: 11 July 2020 / Accepted: 17 July 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract An efficient approximate version of implicit Taylor methods for initial-value problems of systems of ordinary differential equations (ODEs) is introduced. The approach, based on an approximate formulation of Taylor methods, produces a method that requires less evaluations of the function that defines the ODE and its derivatives than the usual version. On the other hand, an efficient numerical solution of the equation that arises from the discretization by means of Newton’s method is introduced for an implicit scheme of any order. Numerical experiments illustrate that the resulting algorithm is simpler to implement and has better performance than its exact counterpart. Keywords Taylor methods · implicit schemes · Explicit schemes · ODE integrators · Approximate formulation Mathematics Subject Classification 65L04 · 65L05 · 65L06
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Pep Mulet [email protected] Antonio Baeza [email protected] Raimund Bürger [email protected] María del Carmen Martí [email protected] David Zorío [email protected]
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Departament de Matemàtiques, Universitat de València, 46100 Burjassot, Spain
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CI2MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
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CI2MA, Universidad de Concepción, Casilla 160-C, Concepción, Chile 0123456789().: V,-vol
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A. Baeza et al.
1 Introduction 1.1 Scope This work is related to numerical methods for the solution of the autonomous system of ordinary differential equations (ODEs): u (t) = f u(t) , t ∈ (t0 , T ], (1.1) T T u(t) = u 1 (t), . . . , u M (t) , f (u) = f 1 (u), . . . , f M (u) , where derivatives of a vector of univariate scalar functions are understood component-wise, posed along with initial data u(t0 ) = u0 . Taylor series methods for the numerical solution of initial-value problems of ODEs compute approximations to the solution of the ODE for the next time instant using a Taylor polynomial of the unknown. The resulting methods are simple, since the expressions required for the iteration are exactly computable (i.e., with no error) from the equation, and the truncation error is governed by the error term of the Taylor formula, so that the order of accuracy of the global error of the method corresponds to the degree of the Taylor polynomial used. However, their implementation depends on the terms involved in the Taylor series, i.e., derivatives of the right-hand side whose computation requires intensive symbolic calculus, and are specific to each individual problem. Moreover, the need for solving auxiliary nonlinear equations, especially within the implicit versions, makes them computationally expensive, especially as the order of accuracy required increases. In this work, we focus o
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