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Lobatto Methods

Introduction

Lobatto methods for the numerical integration of differential equations are named after Rehuel Lobatto. Rehuel Lobatto (1796–1866) was a Dutch mathematician working most of his life as an advisor for the government in the fields of life insurance and of weights and measures. In 1842, he was appointed professor of mathematics at the Royal Academy in Delft (known nowadays as Delft University of Technology). Lobatto methods are characterized by the use of approximations to the solution at the two end points t n and tn+1 of each subinterval of integration [t n , tn+1]. Two well-known Lobatto methods based on the trapezoidal quadrature rule which are often used in practice are the (implicit) trapezoidal rule and the Störmer-Verlet-leapfrog method.

The (Implicit) Trapezoidal Rule

Consider a system of ordinary differential equations (ODEs):

$$\displaystyle{ \dfrac{d} {dt}y = f(t,y) }$$
(1)

where \(f : \mathbb{R} \times \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}\). Starting from y0...

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Jay, L.O. (2015). Lobatto Methods. In: Engquist, B. (eds) Encyclopedia of Applied and Computational Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70529-1_123

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