Isaías Alonso-Mallo

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.

Runge-Kutta methods without order reduction for linear initial boundary value problems

Summary. It is well-known the loss of accuracy when a Runge–Kutta method is used together with the method of lines for the full discretization of an initial boundary value problem. We show that this phenomenon, called order reduction, is caused by wrong boundary values in intermediate stages. With a right choice, the order reduction can be avoided and the optimal order of convergence in time is achieved. We prove this fact for time discretizations of abstract initial boundary value problems based on implicit Runge–Kutta methods. Moreover, we apply these results to the full discretization of parabolic problems by means of Galerkin finite element techniques. We present some numerical examples in order to confirm that the optimal order is actually achieved.

Discrete Absorbing Boundary Conditions for Schrödinger-Type Equations. Construction and Error Analysis

When a partial differential equation in an unbounded domain is solved numerically, it is necessary to introduce artificial boundary conditions. In this paper, a general class of absorbing boundary conditions is constructed for one-dimensional Schrodinger-type equations discretized in space by finite differences. For this, rational approximations to the transparent boundary conditions are used. We study the simplest case in detail, obtaining an estimate for the full discrete error and showing that the discrete problem is weakly unstable. Moreover, we show numerically that the discrete problems associated to higher order absorbing boundary conditions are more unstable. Several numerical experiments confirm the results previously obtained.

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 high order finite element discretization with local absorbing boundary conditions of the linear Schrödinger equation

The goal of this paper is to obtain a high order full discretization of the initial value problem for the linear Schrodinger equation in a finite computational domain. For this we use a high order finite element discretization in space together with an adaptive implementation of local absorbing boundary conditions specifically obtained for linear finite elements, and a high order symplectic time integrator. The numerical results show that it is possible to obtain simultaneously a very good absorption at the boundary and a very small error in the interior of the computational domain.

Discrete absorbing boundary conditions for Schrödinger-type equations: practical implementation

Recently, some absorbing boundary conditions for Schrodingertype equations have been studied by Fevens, Jiang and Alonso-Mallo, and Reguera. These conditions make it possible to obtain a very high absorption at the boundary avoiding the nonlocality of transparent boundary conditions. However, the implementations used in the literature, where the boundary condition is chosen in a manual way in accordance with the solution or fixed independently of the solution, are not practical because of the small absorption. In this paper, a new practical adaptive implementation is developed that allows us to obtain automatically a very high absorption.

Rational methods with optimal order of convergence for partial differential equations

We construct rational approximations to linear, non-homogeneous initial boundary value problems. They are based on A-acceptable rational approximations to the exponential for the time discretization. The order reduction phenomenon is avoided and the optimal order of convergence in time is achieved. The theoretical results are illustrated numerically.

Simulation of coherent structures in nonlinear Schrödinger-type equations

This paper presents some numerical methods to simulate the evolution of coherent structures with small fluctuations, that appear as typical solutions of a class of nonintegrable nonlinear Schrodinger equations. The construction of the methods is particularly focused on two points: on one hand, the generation of the ground state profiles, to be used in the initial data of the simulations, combines a suitable spatial discretization with the resolution of a discrete variational problem. On the other hand, the approximation to leading parameters of these structures is controlled by the time integration. We compare different methods when simulating the evolution of initial ground state profiles and some initial data perturbed from them.

Explicit single step methods with optimal order of convergence for partial differential equations

Abstract We construct explicit single step discretizations of linear, non-homogeneous initial boundary value problems. These methods use polynomial approximations for the time discretization. The order reduction phenomenon, present when a fully discrete Runge–Kutta is used, is avoided and the maximum expected order of convergence is achieved. The results are illustrated numerically.

Time exponential splitting technique for the Klein-Gordon equation with Hagstrom-Warburton high-order absorbing boundary conditions

Klein-Gordon equations on an unbounded domain are considered in one dimensional and two dimensional cases. Numerical computation is reduced to a finite domain by using the Hagstrom-Warburton (H-W) high-order absorbing boundary conditions (ABCs). Time integration is made by means of exponential splitting schemes that are efficient and easy to implement. In this way, it is possible to achieve a negligible error due to the time integration and to study the behavior of the absorption error. Numerical experiments displaying the accuracy of the numerical solution for the two dimensional case are provided. The influence of the dispersion coefficient on the error is also studied.