The Symplectic Splitting Method

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The matrix that propagates the numerical solution can be written as where the entries T1(y) and T2(y) are the Taylor series expansion of cos(y) and sin(y) up to order m, i.e. Notice that The eigenvalues are given by The scheme is stable if The Taylor Method For practical purposes, we require however
The Chebyshev Method The Chebyshev method approximates the action of the exponential on the initial conditions by a near-optimal polynomial given by: where with Here, Tk(x) is the kth Chebyshev polynomial generated from the recursion and T0(x)=1, T1(x)=x. Jk(w) are the Bessel functions of the first kind which provides a superlinear convergence for m > t r(H ) i.e.
Page 1 and 2: Splitting and composition methods f
Page 3 and 4: The Methods Considered I - If the s
Page 5 and 6: The Methods Considered I - If the s
Page 7: The Taylor Method An m-stage Taylor
Page 11 and 12: Taylor order tol=inf tol=10 -8 tol=
Page 13 and 14: The linear time dependent SE with H
Page 15 and 16: The Symplectic Splitting Method The
Page 17 and 18: Error bounds: Splitting Methods: va
Page 19 and 20: LOG(ERROR) 0 -2 -4 -6 -8 -10 -12 Po
Page 21 and 22: LOG(ERROR) 0 -2 -4 -6 -8 -10 -12 Po
Page 23 and 24: References SB, F. Casas, and A. Mur
Page 25 and 26: Splitting methods with complex coef
Page 27 and 28: Splitting methods with complex coef
Page 29 and 30: Splitting for perturbed systems Add
Page 31 and 32: Splitting for perturbed systems Add

The Symplectic Splitting Method

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