S. Pérez-Rodríguez

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.

A family of three-stage third order AMF-W-methods for the time integration of advection diffusion reaction PDEs.

In this paper new three-stage W-methods for the time integration of semi-discretized advection diffusion reaction Partial Differential Equations (PDEs) are provided. In particular, two three-parametric families of W-methods of order three are obtained under a realistic assumption regarding the commutator of the exact Jacobian and the approximation of the Jacobian which defines the corresponding W-method. Specific methods are selected by minimizing error coefficients, enlarging stability regions or increasing monotonicity factors, and embedded methods of order two for an adaptive time integration are derived by further assuming first order approximations to the Jacobian. The relevance of the newly proposed methods in connection with the Approximate Matrix Factorization technique is discussed and numerical illustration on practical PDE problems revealing that the new methods are good competitors over existing integrators in the literature is provided.

AMF-Runge-Kutta formulas and error estimates for the time integration of advection diffusion reaction PDEs

The convergence of a family of AMF-Runge-Kutta methods (in short AMF-RK) for the time integration of evolutionary Partial Differential Equations (PDEs) of Advection Diffusion Reaction type semi-discretized in space is considered. The methods are based on very few inexact Newton Iterations applied to Implicit Runge-Kutta formulas by combining the use of a natural splitting for the underlying Jacobians and the Approximate Matrix Factorization (AMF) technique. This approach allows a very cheap implementation of the Runge-Kutta formula under consideration. Particular AMF-RK methods based on Radau IIA formulas are considered. These methods have given very competitive results when compared with important formulas in the literature for multidimensional systems of non-linear parabolic PDE problems. Uniform bounds for the global time-space errors on semi-linear PDEs when simultaneously the time step-size and the spatial grid resolution tend to zero are derived. Numerical illustrations supporting the theory are presented.

PDE-W-methods for parabolic problems with mixed derivatives

The present work considers the numerical solution of differential equations that are obtained by space discretization (method of lines) of parabolic evolution equations. Main emphasis is put on the presence of mixed derivatives in the elliptic operator. An extension of the alternating-direction-implicit (ADI) approach to this situation is presented. Our stability analysis is based on a scalar test equation that is relevant to the considered class of problems. The novel treatment of mixed derivatives is implemented in third-order W-methods. Numerical experiments and comparisons with standard methods show the efficiency of the new approach. An extension of our treatment of mixed derivatives to 3D and higher dimensional problems is outlined at the end of the article.

W-methods to stabilize standard explicit Runge-Kutta methods in the time integration of advection-diffusion-reaction PDEs

A technique to stabilize standard explicit Runge-Kutta methods by associating them with W-methods is proposed. The main point to get the associated family of W-methods for a given explicit Runge-Kutta method is to require commutativity for the coefficient matrices of the W-method in order to reduce the large number of order conditions that must be satisfied to get a pre-fixed order.Based on this idea, for any given explicit four-stage Runge-Kutta method of order four, two uniparametric families of third order W-methods are obtained. The free parameter can be used to increase the stability regions of the W-methods in case of d ź 1 splittings in the derivative function when a von Neumann stability analysis is carried out. Additionally, it is possible to find L-stable ROW-methods (W-methods with exact Jacobian) for some specific values of the free parameter.The new family of W-methods is also equipped with the splitting provided by the Approximate Matrix Factorization (AMF), which converts a W-method into some kind of ADI-method (Alternating Direction Implicit method). The AMF-W-methods so obtained are mainly used to solve large time-dependent PDE systems (in 2D or 3D spatial variables) discretized in space by using finite differences or finite volumes. Some stability properties of the family of AMF-W-methods are also supplied for the case of d -splittings ( d ź 1 ).Numerical experiments in connection with the proposed AMF-W-methods on a few interesting stiff problems coming from PDE discretizations illustrate the stabilization approach in comparison with some relevant methods in the literature.

A Code Based on Gauss Methods for Second Order Differential Systems

A code for the numerical solution of Initial Value Problems for Second Order Systems y″(t) = f(t,y), is presented. The code, of general scope, can cope satisfactorily with oscillatory problems specially when the low frequencies are dominant and low to medium accuracy is required. It is of interest for medium to large dimensional problems, specially when banded or circulant Jacobian matrices arise in the discretization in space via MoL of some time‐dependant partial differential equations, such as vibrating bars. The code is equipped with a reliable global error estimate and it is based on the two‐stage Runge‐Kutta Gauss method. The stage values are solved by an special Newton‐type iteration and the predictors were carefully chosen to minimize the number of iterations per integration step. A guess for the initial step‐size is provided and a variable step‐size policy is used. A continuously differentiable solution based on the Hermite interpolation is supplied. The performance of the code on some interestin...