Approximate Taylor methods for ODEs

On a Class of Conjugate Symplectic Hermite-Obreshkov One-Step Methods with Continuous Spline Extension

“…Clearly, there is a price for this because total derivatives of the f function are involved in the difference equation defining the method, and thus, a suitable smoothness requirement for f is necessary. Multiderivative methods have been considered often in the past for the numerical treatment of ODEs, for example also in the context of boundary value methods [2], and in the last years, there has been a renewed interest in this topic, also considering its application to the numerical solution of differential algebraic equations; see, e.g., [3][4][5][6][7][8]. Here, we consider the numerical solution of Hamiltonian problems which in canonical form can be written as follows:…”

On a Class of Hermite-Obrechkoff One-Step Methods with Continuous Spline Extension

“…Referring to the methods obtainable by Lemma 1, if in particular the polynomial P 2R is defined as in (8), we obtain the class of methods we are here interested in. They can be written as in (6)…”

“…Multiderivative methods have been considered often in the past for the numerical treatment of ordinary differential equations, for example also in the context of boundary value methods [4] and in the last years there is a renewed interest in this topic, also considering its application to the numerical solution of differential algebraic equations, see e.g. [5][6][7][8][9][10]. Here we consider the numerical solution of Hamiltonian problems which in canonical form can be written as follows y = J∇H(y), y(t 0 ) = y 0 ∈ IR 2 ,…”

“…R, exists if and only if(6) is fulfilled with the β coefficients defined as in(9). The piecewise polynomial whose restriction to [t n , t n+1 ] coincides with this polynomial is clearly a C R spline of degree 2R with breakpoints at the mesh points and it collocates the differential equation at the mesh points with multiplicity R 2.…”

On approximate implicit Taylor methods for ordinary differential equations

“…In this work, we focus on implicit Taylor methods, obtained by computing the Taylor polynomials centered on a future time instant, and often used to solve problems where explicit methods have strong stability restrictions, in particular stiff systems of ODEs (Hairer and Wanner 1996). First of all, we apply a strategy, based on the work by Baeza et al (2017) for the explicit Taylor method, to efficiently approximate the derivatives of f . This approximation inherits the ease of implementation and performance of the explicit version.…”