N. Kopell. Invariant manifolds and the initialization problem for some atmospheric equations. Phys. D, 14(2):203-- 215, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....method [11] However, Assumption 2 is not satisfied for singularly perturbed Hamiltonian systems. In particular, solutions of (1) oscillate highly about the manifold M 0 . Thus, as we will show in Section 2, the manifold M 0 does not even satisfy the weaker assumption of normal hyperbolicity [4] [6]. This leaves us with the task of finding a different approach to the long time integration of (1) In this paper, we attempt to do so by introducing the notion of the smoothed dynamics of highly oscillatory Hamiltonian systems. By this we mean the following: Because of (3) the shortest period in ....

....manifold M 0 is given by q 1 = p 1 = 0 or, in the original variables by (11) Linearization of (16) about the manifold M 0 yields a linear system with eigenvalues on the imaginary axis. Thus the manifold M 0 is not normally hyperbolic and the persistence of M o for ffl 0 cannot be concluded [6]. Let us now review a few results from statistical mechanics. Under the assumption that a given Hamiltonian system is ergodic, equipartition of energy [12] implies that hp i H p i i = ffi (17) and hq i H q i i = Gammaffi (18) where q i and p i , i = 1; n, denote the ith ....

Kopell, N., Invariant manifolds and the initialization problem for some atmospheric equations, Physica D, 14, 203--215, 1985.

....method [15] However, Assumption 2 is not satisfied for singularly perturbed Hamiltonian systems. In particular, solutions of (1) oscillate highly about the manifold M 0 . Thus, as we will show in Section 2, the manifold M 0 does not even satisfy the weaker assumption of normal hyperbolicity [6] [8]. This leaves us with the task of finding a different approach to the long time integration of (1) In this paper, we attempt to do so by introducing the notion of the smoothed dynamics of highly oscillatory Hamiltonian systems. By this we mean the following: Because of (4) the shortest period in ....

....manifold M 0 is given by q 1 = p 1 = 0 or, in the original variables by (14) Linearization of (18) about the manifold M 0 yields a linear system with eigenvalues on the imaginary axis. Thus the manifold M 0 is not normally hyperbolic and the persistence of M o for ffl 0 cannot be concluded [8]. Now we want to derive a few important properties of the smoothing operator (9) We assume that ae : IR IR is a smooth function that goes to zero, as jtj 1, faster than any inverse power of t, ae(0) 1, and Z 1 Gamma1 ae(t)dt = 1 The proper construction of a ae, such that in addition ....

Kopell, N., Invariant manifolds and the initialization problem for some atmospheric equations, Physica D, 14, 203--215, 1985.

.... Runge Kutta methods of stiff mechanical systems of a strong potential energy, e.g. stiff spring force such as the stiff pendulum (5) converges to the slowly varying part of the solution, with the stepsize independent of the parameter ffl in (5) Reich [22] extends the principle of slow manifold [9, 15] to DAE of MBS with highly oscillatory force terms. Algebraic constraints corresponding to the slow motion were introduced with a relaxation parameter to preserve the slow solution while adding flexibility to it in the slow manifold approach. 0 1 2 3 4 5 20 10 0 10 20 40 20 0 20 40 Time ....

N. Kopell, Invariant manifolds and initialization problem for some atmospheric equations, Physica D., (14), 203-215, 1985.

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N. Kopell. Invariant manifolds and the initialization problem for some atmospheric equations. Phys. D, 14(2):203-- 215, 1985.

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