A Concise Introduction to Geometric Numerical Integration - download pdf or read online

By Sergio Blanes, Fernando Casas

ISBN-10: 1482263424

ISBN-13: 9781482263428

ISBN-10: 1482263440

ISBN-13: 9781482263442

Discover How Geometric Integrators protect the most Qualitative homes of continuing Dynamical Systems

A Concise creation to Geometric Numerical Integration offers the most issues, concepts, and functions of geometric integrators for researchers in arithmetic, physics, astronomy, and chemistry who're already conversant in numerical instruments for fixing differential equations. It additionally bargains a bridge from conventional education within the numerical research of differential equations to knowing contemporary, complicated examine literature on numerical geometric integration.

The e-book first examines high-order classical integration tools from the constitution maintenance standpoint. It then illustrates the best way to build high-order integrators through the composition of uncomplicated low-order tools and analyzes the assumption of splitting. It subsequent reports symplectic integrators developed without delay from the idea of producing features in addition to the real class of variational integrators. The authors additionally clarify the connection among the maintenance of the geometric homes of a numerical approach and the saw favorable blunders propagation in long-time integration. The e-book concludes with an research of the applicability of splitting and composition how you can definite periods of partial differential equations, similar to the Schrödinger equation and different evolution equations.

The motivation of geometric numerical integration isn't just to boost numerical equipment with more desirable qualitative habit but in addition to supply extra actual long-time integration effects than these received via general-purpose algorithms. obtainable to researchers and post-graduate scholars from varied backgrounds, this introductory ebook will get readers on top of things at the principles, equipment, and purposes of this box. Readers can reproduce the figures and effects given within the textual content utilizing the MATLAB® courses and version documents on hand online.

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Extra resources for A Concise Introduction to Geometric Numerical Integration

Example text

2 Stability Another important concept associated with the numerical treatment of ordinary differential equations is stability. Intuitively, a method is stable if the corresponding numerical solution is bounded when the exact solution does not tend to infinity. In other words, the method does not magnify the differences xn − x(tn ) as tn increases, irrespective of the time step at which the error is produced. There are several ways to express this stability condition. 24) with λ > 0, and initial condition x(t0 ) = x0 .

M, to be used as follows: [q p] = phiKepler(q0,p0,t,mu). The exact solution obtained in this way can be used for comparison and testing of different numerical schemes, just by measuring the error in coordinates and momenta along the time integration. Since we are dealing with a central force, the motion takes place in a plane and so we may analyze it as a two-dimensional problem. 61) if 0 ≤ e < 1 the total energy is H = H0 = −1/2, the solution is periodic with period 2π and the trajectory is an ellipse of eccentricity e.

42) verifies N T JN = J. 41) is symplectic. 8. Verify that the Jacobi identity is indeed satisfied for any functions F (q, p), G(q, p) and H(q, p) of the canonical variables q, p. 9. Given the functions T (p) and V (q) in one dimension, compute the following Poisson brackets: (i) {V (q), T (p)}, (ii) {V (q), {V (q), T (p)}}, (iii) {T (p), {V (q), T (p)}}. Repeat the computation for any d > 1. 10. Given the quadratic function T (p) = 21 pT M −1 p, where M is a positive definite matrix, and a general function V (q), compute the following Poisson brackets: (i) {V (q), T (p)}, (ii) {V (q), {V (q), T (p)}}, (iii) {V (q), {V (q), {V (q), T (p)}}}.

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A Concise Introduction to Geometric Numerical Integration by Sergio Blanes, Fernando Casas

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