B. Cano

Spectral-fractional step Runge–Kutta discretizations for initial boundary value problems with time dependent boundary conditions

In this paper we develop a technique for avoiding the order reduction caused by nonconstant boundary conditions in the methods called splitting, alternating direction or, more generally, fractional step methods. Such methods can be viewed as the combination of a semidiscrete in time procedure with a special type of additive Runge–Kutta method, which is called the fractional step Runge–Kutta method, and a standard space discretization which can be of type finite differences, finite elements or spectral methods among others. Spectral methods have been chosen here to complete the analysis of convergence of a totally discrete scheme of this type of improved fractionary steps. The numerical experiences performed also show the increase of accuracy that this technique provides.

Spectral/Rosenbrock discretizations without order reduction for linear parabolic problems

The order reduction phenomenon occurs when a Rosenbrock method is used together with the method of lines for the full discretization of an initial boundary value problem. This phenomenon can be avoided with a right choice of the boundary values of the intermediate stages. This fact is proved for time discretizations of abstract initial boundary value problems with variable stepsize. These results are applied for the study of full discretizations of parabolic problems by using spectral methods for the spatial discretization. Some numerical examples confirm that the optimal order is achieved.

A comparison of symplectic and Hamilton's principle algorithms for autonomous and non-autonomous systems of ordinary differential equations

After a thorough study of order and symmetry of variational methods based on Hamiltons principle, we study efficient implementations that make them competitive with symplectic methods for some of our test problems.

A generalization to variable stepsizes of Sto¨rmer methods for second-order differential equations

Abstract A family of methods that extend Stormer formulae to variable stepsizes is introduced, analyzed and tested. The variable stepsize methods do not satisfy the generalized root condition that in their fixed stepsize counterparts is equivalent to stability, but are shown to be stable and convergent. However, numerical experience suggests that they cannot compete with modern embedded pairs of Runge-Kutta-Nystrom methods.

Stable Runge-Kutta-Nyström methods for dissipative stiff problems

The definition of stability for Runge–Kutta–Nyström methods applied to stiff second-order in time problems has been recently revised, proving that it is necessary to add a new condition on the coefficients in order to guarantee the stability. In this paper, we study the case of second-order in time problems in the nonconservative case. For this, we construct an

Avoiding order reduction when integrating nonlinear Schrödinger equation with Strang method

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Stability of Runge-Kutta-Nyström methods

-stable Runge–Kutta–Nyström method with two stages satisfying this condition of stability and we show numerically the advantages of this new method.

Multistep cosine methods for second-order partial differential systems

In this paper a technique is suggested to avoid order reduction when using Strang method to integrate nonlinear Schrodinger equation subject to time-dependent Dirichlet boundary conditions. The computational cost of this technique is negligible compared to that of the method itself, at least when the timestepsize is fixed. Moreover, a thorough error analysis is given as well as a modification of the technique which allows to conserve the symmetry of the method while retaining its second order.