By Ernst Hairer, Christian Lubich, Gerhard Wanner
This e-book covers numerical equipment that protect homes of Hamiltonian structures, reversible structures, differential equations on manifolds and issues of hugely oscillatory ideas.
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Extra info for Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations
Example text
3. 8) is the StOrmerNerlet method. 5 Splitting Methods The splitting idea yields an approach that is completely different from Runge-Kutta methods. One decomposes the vector field into integrable pieces and treats them separately. 42 II. tllllll! t tflll ///,ftf//I //////// ! 11 t f T t t t t t t t //////// //////// t t t t t Fig. 1. A splitting of a vector field. We consider an arbitrary system field is "split" as (see Fig. 1) If then, by chance, the exact flows 'P~I] and 'P~2] of the systems iJ = 1[1] (y) and iJ = 1[2] (y) can be calculated explicitly, we can, from a given initial value Yo, first solve the first system to obtain a value Yl/2, and from this value integrate the second system to obtain Yl.
A problem y = f(t, y) can be brought into this form by appending the equation i = 1. We develop the subsequent theory in four steps. E. , Geometric Numerical Integration © Springer-Verlag Berlin Heidelberg 2002 48 III. Order Conditions, Trees and B-Series Er sagte es klar und angenehm, was erstens, zweitens und drittens kfun'. (W. Busch, lobsiade 1872) First Step. We compute the higher derivatives of the solution y at the initial point to. 2) and compute the latter derivatives by using the chain rule, the product rule, the symmetry of partial derivatives, and the notation f' (y) for the derivative as a linear map (the Jacobian), f" (y) the second derivative as a bilinear map and similarly for higher derivatives.
1), but the StormerNerlet scheme does as well. Implementation of the StarmerlVerlet scheme. 4). Runge-Lenz-Pauli vector. Prove that the function q2 7. 8. 9. 10. A(p, q) = (~~) x ( o ~ qlP2 - q2Pl ) 22 I. , A (p( t), q( t)) = Canst along solutions of the problem. However, it is not a first integral of the perturbed Kepler problem of Exercise 12. 11. 1 which shows the long-time behaviour of the error in the Runge-Lenz-Pauli vector (see Exercise 10) for the various numerical integrators. 12. Study numerically the solution of the perturbed Kepler problem with Hamiltoman where JL is a positive or negative small number.