• openAccess   An efficient algorithm based on splitting for the time integration of the Schrödinger equation 

      Blanes, Sergio; Casas, Fernando; Murua, Ander Elsevier (2015-12)
      We present a practical algorithm based on symplectic splitting methods intended for the numerical integration in time of the Schrödinger equation when the Hamiltonian operator is either time-independent or changes slowly ...
    • openAccess   An efficient algorithm for computing the Baker–Campbell–Hausdorff series and some of its applications 

      Casas, Fernando; Murua, Ander American Institute of Physics (2009-03)
      We provide a new algorithm for generating the Baker–Campbell–Hausdorff (BCH) seriesZ = log(eXeY) in an arbitrary generalized Hall basis of the free Lie algebra L(X,Y)generated by X and Y. It is based on the close relationship ...
    • openAccess   Continuous changes of variables and the Magnus expansion 

      Casas, Fernando; Chartier, Philippe; Murua, Ander IOP (2019-09-23)
      In this paper, we are concerned with a formulation of Magnus and Floquet-Magnus expansions for general nonlinear differential equations. To this aim, we introduce suitable continuous variable transformations generated by ...
    • openAccess   Efficient computation of the Zassenhaus formula 

      Casas, Fernando; Murua, Ander; Nadinic, Mladen Elsevier (2012-11)
      A new recursive procedure to compute the Zassenhaus formula up to high order is presented, providing each exponent in the factorization directly as a linear combination of independent commutators and thus containing the ...
    • openAccess   Error analysis of splitting methods for the time dependent Schrödinger equation 

      Blanes, Sergio; Casas, Fernando; Murua, Ander Society for Industrial and Applied Mathematics (2011)
      A typical procedure to integrate numerically the time dependent Schrödinger equation involves two stages. In the first stage one carries out a space discretization of the continuous problem. This results in the linear ...
    • openAccess   High precision Symplectic Integrators for the Solar System 

      Farrés Basiana, Ariadna; Laskar, Jacques; Blanes, Sergio; Casas, Fernando; Makazaga, Joseba; Murua, Ander Springer Netherlands (2013-02)
      Using a Newtonian model of the Solar System with all 8 planets, we perform extensive tests on various symplectic integrators of high orders, searching for the best splitting scheme for long term studies in the Solar System. ...
    • openAccess   New families of symplectic splitting methods for numerical integration in dynamical astronomy 

      Blanes, Sergio; Casas, Fernando; Farrés Basiana, Ariadna; Laskar, Jacques; Makazaga, Joseba; Murua, Ander Elsevier (2013-06)
      We present new splitting methods designed for the numerical integration of near-integrable Hamiltonian systems, and in particular for planetary N-body problems, when one is interested in very accurate results over a large ...
    • openAccess   On the Linear Stability of Splitting Methods 

      Blanes, Sergio; Casas, Fernando; Murua, Ander Springer Verlag (2008)
      A comprehensive linear stability analysis of splitting methods is carried out by means of a 2 × 2 matrix K(x) with polynomial entries (the stability matrix) and the stability polynomial p(x) (the trace of K(x) divided by ...
    • openAccess   Optimized high-order splitting methods for some classes of parabolic equation 

      Blanes, Sergio; Casas, Fernando; Chartier, Philippe; Murua, Ander American Mathematical Society (2013)
      Weareconcernedwiththenumericalsolutionobtainedbysplitting methods of certain parabolic partial differential equations. Splitting schemes of order higher than two with real coefficients necessarily involve negative coefficients. ...
    • openAccess   Splitting and composition methods in the numerical integration of differential equations 

      Blanes, Sergio; Murua, Ander; Casas, Fernando Sociedad Española de Matemática Aplicada (2008)
      We provide a comprehensive survey of splitting and composition methods for the numerical integration of ordinary differential equations (ODEs). Splitting methods constitute an appropriate choice when the vector field ...
    • openAccess   Splitting methods in the numerical integration of non-autonomous dynamical systems 

      Casas, Fernando; Blanes, Sergio; Murua, Ander Springer (2012)
      We present a procedure leading to efficient splitting schemes for the time integration of explicitly time dependent partitioned linear differential equa- tions arising when certain partial differential equations are ...
    • openAccess   Splitting methods with complex coefficients 

      Blanes, Sergio; Casas, Fernando; Murua, Ander Sociedad Española de Matemática Aplicada (2010)
      Splitting methods for the numerical integration of differential equations of order greater than two involve necessarily negative coefficients. This order barrier can be overcome by considering complex coefficients with ...
    • openAccess   Symplectic time-average propagators for the Schrodinger equation with a time-dependent Hamiltonian 

      Blanes, Sergio; Casas, Fernando; Murua, Ander AIP Publishing (2017-03)
      Several symplectic splitting methods of orders four and six are presented for the step-by-step time numerical integration of the Schrödinger equation when the Hamiltonian is a general explicitly time-dependent real operator. ...