structure preserving algorithms with adaptive time step are systematically developed for Birkhoffian systems. The development mainly consists of construction, implementation, and application of this kind of algorithms...
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structure preserving algorithms with adaptive time step are systematically developed for Birkhoffian systems. The development mainly consists of construction, implementation, and application of this kind of algorithms. The construction is based on a direct discretization of the Pfaff-Birkhoff principle in which time is treated as a dynamical variable particularly. The resulting discrete Birkhoffian equations then determine a numerical algorithm for iteration which automatically meets the requirement of structure preservation and time step adaptation. Following the construction, an alternative optimization technique of solving discrete Birkhoffian equations and a reasonable method for initialization of the simulation are provided subsequently, for practical implementation of the algorithm. The performance of the developed algorithm is examined finally by preliminary application in typical examples. Numerical results indicate that the time step adaptation leads to a much improved performance on precisely preserving conserved quantities.
The Bethe-Salpeter eigenvalue problem is a dense structured eigenvalue problem arising from discretized Bethe-Salpeter equation in the context of computing exciton energies and states. A computational challenge is tha...
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The Bethe-Salpeter eigenvalue problem is a dense structured eigenvalue problem arising from discretized Bethe-Salpeter equation in the context of computing exciton energies and states. A computational challenge is that at least half of the eigenvalues and the associated eigenvectors are desired in practice. We establish the equivalence between Bethe-Salpeter eigenvalue problems and real Hamiltonian eigenvalue problems. Based on theoretical analysis, structure preserving algorithms for a class of Bethe-Salpeter eigenvalue problems are proposed. We also show that for this class of problems all eigenvalues obtained from the Tamm-Dancoff approximation are overestimated. In order to solve large scale problems of practical interest, we discuss parallel implementations of our algorithms targeting distributed memory systems. Several numerical examples are presented to demonstrate the efficiency and accuracy of our algorithms. (C) 2015 Elsevier Inc. All rights reserved.
In this paper the issue of integrating matrix differential systems whose solutions are unitary matrices is addressed. Such systems have skew-Hermitian coefficient matrices in the linear case and a related structure in...
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In this paper the issue of integrating matrix differential systems whose solutions are unitary matrices is addressed. Such systems have skew-Hermitian coefficient matrices in the linear case and a related structure in the nonlinear case. These skew systems arise in a number of applications, and interest originates from application to continuous orthogonal decoupling techniques. In this case, the matrix system has a cubic nonlinearity. Numerical integration schemes that compute a unitary approximate solution for all stepsizes are studied. These schemes can be characterized as being of two classes: automatic and projected unitary schemes. In the former class, there belong those standard finite difference schemes which give a unitary solution;the only ones are in fact the Gauss-Legendre point Runge-Kutta (Gauss RK) schemes. The second class of schemes is created by projecting approximations computed by an arbitrary scheme into the set of unitary matrices. In the analysis of these unitary schemes, the stability considerations are guided by the skew-Hermitian character of the problem. Various error and implementation issues are considered, and the methods are tested on a number of examples.
The numerical solution of non-canonical Hamiltonian systems is an active and still growing field of research. At the present time, the biggest challenges concern the realization of structure preserving algorithms for ...
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The numerical solution of non-canonical Hamiltonian systems is an active and still growing field of research. At the present time, the biggest challenges concern the realization of structure preserving algorithms for differential equations on infinite dimensional manifolds. Several classical PDEs can indeed be set in this framework. In this thesis, I develop a new class of numerical schemes for Hamiltonian isospectral flows, in order to solve the hydrodynamical Euler equations on a sphere. The results are presented in two papers. In the first one, we derive a general framework for the isospectral flows, providing then a class of numerical methods of arbitrary order, based on the Lie–Poisson reduction of Hamiltonian systems. Avoiding the use of any constraint, weobtain a large class of numerical schemes for Hamil- tonian and non-Hamiltonian isospectral flows. One of the advantages of these methods is that, together with the isospectrality, they have near conservation of the Hamiltonian and, indeed, they are Lie–Poisson inte- grators. In the second paper, using the results of the first one, we present a numerical method based on the geometric quantization of the Poisson algebra of the smooth functions on a sphere, which gives an approximate solution of the Euler equations with a number of discrete first integrals which is consistent with the level of discretization.
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