We analyze composition methods with complex coefficients exhibiting the so-called “symmetry-conjugate” pattern in their distribution. In particular, we study their behavior with respect to preservation of qualitative properties when projected on the real axis and we compare them with the usual left-right palindromic compositions. New schemes within this family up to order 8 are proposed and their efficiency is tested on several examples. Our analysis shows that higher-order schemes are more efficient even when time step sizes are relatively large.
There is increasing numerical evidence that scalar fields can form long-lived quasi-bound states around black holes. Recent perturbative and numerical relativity calculations have provided further confirmation in a variety of physical systems, including both static and accreting black holes, and collapsing fermionic stars. In this work we investigate this issue yet again in the context of gravitationally unstable boson stars leading to black hole formation. We build a large sample of spherically symmetric initial models, both stable and unstable, incorporating a self-interaction potential with a quartic term. The three different outcomes of unstable models, namely migration to the stable branch, total dispersion, and collapse to a black hole, are also present for self-interacting boson stars. Our simulations show that for black-hole-forming models, a scalar-field remnant is found outside the black-hole horizon, oscillating at a different frequency than that of the original boson star. This result is in good agreement with recent spherically symmetric simulations of unstable Proca stars collapsing to black holes [1].
<p style='text-indent:20px;'>We explore the applicability of splitting methods involving complex coefficients to solve numerically the time-dependent Schrödinger equation. We prove that a particular class of integrators are conjugate to unitary methods for sufficiently small step sizes when applied to problems defined in the group <inline-formula><tex-math id="M1">\begin{document}$ \mathrm{SU}(2) $\end{document}</tex-math></inline-formula>. In the general case, the error in both the energy and the norm of the numerical approximation provided by these methods does not possess a secular component over long time intervals, when combined with pseudo-spectral discretization techniques in space.</p>
New families of fourth-order composition methods for the numerical integration of initial value problems defined by ordinary differential equations are proposed. They are designed when the problem can be separated into three parts in such a way that each part is explicitly solvable. The methods are obtained by applying different optimization criteria and preserve geometric properties of the continuous problem by construction. Different numerical examples exhibit their improved performance with respect to previous splitting methods in the literature. used in GNI because the composition of symplectic (or volume preserving, orthogonal, etc.) transformations is again symplectic (volume preserving, orthogonal, etc., respectively). In composition methods the numerical scheme is constructed as the composition of several simpler integrators for the problem at hand, so as to improve their accuracy.When in (1) can be separated into two parts, very efficient splitting schemes have been designed and applied to solve a wide variety of problems arising in several fields, ranging from Hamiltonian Monte Carlo techniques to the evolution of the -body gravitational problem in Celestial Mechanics (see [3,4] and references therein).There are, however, relevant problems in applications where has to be decomposed into three or more parts in order to have subproblems that are explicitly solvable. Examples include the disordered discrete nonlinear Schrödinger equation [7], Vlasov-Maxwell equations in plasma physics [8], the motion of a charged particle in an electromagnetic field according with the Lorentz force law [9] and problems in molecular dynamics [10]. In that case, although in principle methods of any order of accuracy can be built, the resulting algorithms involve such a large number of maps that they are not competitive in practice. It is the purpose of this paper to present an alternative class of efficient methods for the problem at hand and compare their performance on some non-trivial physical examples than can be split into three parts.The paper is structured as follows. We first review how splitting methods can be directly applied to get numerical solutions (Section 2). Then the attention is turned to the application of composition methods, and we get a family of 4th-order schemes obtained by applying a standard optimization procedure (Section 3). In Section 4 we show how standard splitting methods, when formulated as a composition scheme, lead to very competitive integrators, and also propose a different optimization criterion for systems possessing invariant quantities. This allows us to get a new family of 4th-order schemes. All these integration algorithms are subsequently tested in Section 5 on a pair of numerical examples. Finally, Section 6 contains some concluding remarks.
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