S. González-Pinto

Iterative schemes for three-stage implicit Runge-Kutta methods

Abstract In this paper we consider a class of iterative schemes for implicit Runge-Kutta methods. Taking into account convergence and linear stability properties, we propose a technique to develop efficient schemes for three-stage methods and we construct particular schemes for the case of Gauss and RadauII A methods. Finally, some numerical experiments are included in order to show the efficiency of the algorithms.

Iterative schemes for Gauss methods

Abstract In this paper, we consider a class of iterative schemes for implicit Runge-Kutta methods and we study the convergence of these schemes for a family of nonlinear stiff problems. A particular convergent scheme for the two stage Gauss method is proposed and the order and linear stability properties are analyzed. Finally, some numerical experiments are included in order to show the efficiency of the method.

Implementation of high-order implicit runge-kutta methods

Abstract In this paper, we develop single-Newton iterative schemes for the solution of the stage equations of some implicit Runge-Kutta methods such as the four-stage Gauss and Radau IIA methods and the five-stage Lobatto IIIA formula. We also compare the implementation cost of these schemes with the simplified-Newton iteration and we present some numerical experiments on some well-known stiff test problems to show that the proposed iterations are reliable and efficient.

On the Starting Algorithms for Fully Implicit Runge-Kutta Methods

This paper is concerned with the behavior of starting algorithms to solve the algebraic equations of stages arising when fully implicit Runge-Kutta methods are applied to stiff initial value problems. The classical Lagrange extrapolation of the internal stages of the preceding step and some variants thereof that do not require any additional cost are analyzed. To study the order of the starting algorithms we consider three different approaches. First we analyze the classical order through the theory of Butchers series, second we derive the order on the Prothero and Robinson model and finally we study the stiff order for a general class of dissipative problems. A detailed study of the orders of some starting algorithms for Gauss, Radau IA-IIA, Lobatto IIIA-C methods is also carried out. Finally, to compare the most relevant starting algorithms studied here, some numerical experiments on well known nonlinear stiff problems are presented.

Variable-order starting algorithms for implicit Runge-Kutta methods on stiff problems

This paper deals with starting algorithms for Newton-type schemes for solving the stage equations of implicit s-stages Runge-Kutta methods applied to stiff problems. We present a family of starting algorithms with orders from 0 to s + 1 and, with estimations of the error in these algorithms, we give a technique for selecting, at each step, the most convenient in the family. The proposed algorithms, that can be expressed in terms of divided differences, are based on the Lagrange interpolation of the stages of the last two integration steps. We also analyse the orders of the starting algorithms for the non-stiff case, for the Prothero and Robinson model and the stiff order. Finally, by means of some numerical experiments we show that this technique allows, in general, to greatly improve the performance of implicit Runge-Kutta methods on stiff problems.

Runge-Kutta Methods for the Numerical Solution of Stiff Semilinear Systems

This paper studies the stability and convergence properties of general Runge-Kutta methods when they are applied to stiff semilinear systems y′(t) = J(t)y(t) + g(t, y(t)) with the stiffness contained in the variable coefficient linear part.We consider two assumptions on the relative variation of the matrix J(t) and show that for each of them there is a family of implicit Runge-Kutta methods that is suitable for the numerical integration of the corresponding stiff semilinear systems, i.e. the methods of the family are stable, convergent and the stage equations possess a unique solution. The conditions on the coefficients of a method to belong to these families turn out to be essentially weaker than the usual algebraic stability condition which appears in connection with the B-stability and convergence for stiff nonlinear systems. Thus there are important RK methods which are not algebraically stable but, according to our theory, they are suitable for the numerical integration of semilinear problems.This paper also extends previous results of Burrage, Hundsdorfer and Verwer on the optimal convergence of implicit Runge-Kutta methods for stiff semilinear systems with a constant coefficients linear part.

A comparison of AMF- and Krylov-methods in Matlab for large stiff ODE systems

For the efficient solution of large stiff systems resulting from semidiscretization of multi-dimensional partial differential equations two methods using approximate matrix factorizations (AMF) are discussed. In extensive numerical tests of Reaction Diffusion type implemented in Matlab they are compared with integration methods using Krylov techniques for solving the linear systems or to approximate exponential matrices times a vector. The results show that for low and medium accuracy requirements AMF methods are superior. For stringent tolerances peer methods with Krylov are more efficient.

Two-step error estimators for implicit Runge--Kutta methods applied to stiff systems

This paper is concerned with local error estimation in the numerical integration of stiff systems of ordinary differential equations by means of Runge--Kutta methods. With implicit Runge--Kutta methods it is often difficult to embed a local error estimate with the appropriate order and stability properties. In this paper local error estimation based on the information from the last two integration steps (that are supposed to have the same steplength) is proposed. It is shown that this technique, applied to Radau IIA methods, lets us get estimators with proper order and stability properties. Numerical examples showing that the proposed estimate improves the efficiency of the integration codes are presented.

Stabilized starting algorithms for collocation Runge-Kutta methods☆

Abstract In this paper, we propose a technique to stabilize some starting algorithms often used in the Newton-type iterations appearing when collocation Runge-Kutta methods are applied to solve stiff initial value problems. By following the ideas given in [1], we analyze the order (classical and stiff) of the new starting algorithms and pay special attention to their error amplifying functions. From the computational point of view, the new algorithms require the solution of an additional linear system per integration step, but as shown in the numerical experiments, this extra cost is compensated in most of the problems by their better stability properties.

A numerical scheme to approximate the solution of a singularly perturbed nonlinear differential equation

Abstract In this paper an “analytic” difference scheme to integrate the singularly perturbed scalar problem (1.1) is given. The algorithm is based on the exact integration of a locally linearized problem (on a special nonuniform mesh) exhibiting uniform convergence in ϵ (for any x ). The scheme is exact when f = a + bx + cy , where a , b , c are constants. Finally numerical results are given.