0000000000417087
AUTHOR
Sergio Blanes
showing 6 related works from this author
High-order Runge–Kutta–Nyström geometric methods with processing
2001
Abstract We present new families of sixth- and eighth-order Runge–Kutta–Nystrom geometric integrators with processing for ordinary differential equations. Both the processor and the kernel are composed of explicitly computable flows associated with non trivial elements belonging to the Lie algebra involved in the problem. Their efficiency is found to be superior to other previously known algorithms of equivalent order, in some case up to four orders of magnitude.
Continuous numerical solutions of coupled mixed partial differential systems using Fer's factorization
1999
In this paper continuous numerical solutions expressed in terms of matrix exponentials are constructed to approximate time-dependent systems of the type ut A(t)uxx B(t)u=0; 0 0, u(0;t)=u(p;t)=0; u(x;0)=f(x);06 x6p. After truncation of an exact series solution, the numerical solution is constructed using Fer’s factorization. Given >0 and t0;t1; with 0<t0<t1 and D(t0;t1)=f(x;t); 06x6p; t06t6t1g the error of the approximated solution with respect to the exact series solution is less than uniformly in D(t0;t1). An algorithm is also included. c 1999 Elsevier Science B.V. All rights reserved. AMS classication: 65M15, 34A50, 35C10, 35A50
New Families of Symplectic Runge-Kutta-Nyström Integration Methods
2001
We present new 6-th and 8-th order explicit symplectic Runge-Kutta-Nystrom methods for Hamiltonian systems which are more efficient than other previously known algorithms. The methods use the processing technique and non-trivial flows associated with different elements of the Lie algebra involved in the problem. Both the processor and the kernel are compositions of explicitly computable maps.
The Magnus expansion and some of its applications
2008
Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem, shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an elegant setting to build up approximate exponential representations of the solution of the system. It provides a power series expansion for the corresponding exponent and is sometimes referred to as Time-Dependent Exponential Perturbation Theory. Every Magnus approximant corresponds in Perturbation Theory to a partial re-summation of infinite terms with the important additional property of prese…
A pedagogical approach to the Magnus expansion
2010
Time-dependent perturbation theory as a tool to compute approximate solutions of the Schrodinger equation does not preserve unitarity. Here we present, in a simple way, how the Magnus expansion (also known as exponential perturbation theory) provides such unitary approximate solutions. The purpose is to illustrate the importance and consequences of such a property. We suggest that the Magnus expansion may be introduced to students in advanced courses of quantum mechanics.
Magnus and Fer expansions for matrix differential equations: the convergence problem
1998
Approximate solutions of matrix linear differential equations by matrix exponentials are considered. In particular, the convergence issue of Magnus and Fer expansions is treated. Upper bounds for the convergence radius in terms of the norm of the defining matrix of the system are obtained. The very few previously published bounds are improved. Bounds to the error of approximate solutions are also reported. All results are based just on algebraic manipulations of the recursive relation of the expansion generators.