Open Access
Avoiding the order reduction when solving second-order in time PDEs with Fractional Step Runge–Kutta–Nyström methods
(Elsevier, 2016) Moreta, M. Jesús; Bujanda Cirauqui, Blanca; Jorge Ulecia, Juan Carlos; Ingeniería Matemática e Informática; Matematika eta Informatika Ingeniaritza
We study some of the main features of Fractional Step Runge–Kutta–Nyström methods when they are used to integrate Initial–Boundary Value Problems of second order in time, in combination with a suitable spatial discretization. We focus our attention on the order reduction phenomenon, which appears if classical boundary conditions are taken at the internal stages. This drawback is specially hard when time dependent boundary conditions are considered. In this paper we present an efficient technique, very simple and computationally cheap, which allows us to avoid the order reduction; such technique consists in modifying the boundary conditions for the internal stages of the method.
Open Access
A combined fractional step domain decomposition method for the numerical integration of parabolic problems
(Springer, 2004) Portero Egea, Laura; Bujanda Cirauqui, Blanca; Jorge Ulecia, Juan Carlos; Ingeniería Matemática e Informática; Matematika eta Informatika Ingeniaritza
In this paper we develop parallel numerical algorithms to solve linear time dependent coefficient parabolic problems. Such methods are obtained by means of two consecutive discretization procedures. Firstly, we realize a time integration of the original problem using a Fractional Step Runge Kutta method which provides a family of elliptic boundary value problems on certain subdomains of the original domain. Next, we discretize those elliptic problems by means of standard techniques. Using this framework, the numerical solution is obtained by solving, at each stage, a set of uncoupled linear systems of low dimension. Comparing these algorithms with the classical domain decomposition methods for parabolic problems, we obtain a reduction of computational cost because of, in this case, no Schwarz iterations are required. We give an unconditional convergence result for the totally discrete scheme and we include two numerical examples that show the behaviour of the proposed method.
Open Access
An efficient numerical method for singularly perturbed time dependent parabolic 2D convection-diffusion systems
(Elsevier, 2018) Clavero, Carmelo; Jorge Ulecia, Juan Carlos; Ingeniería Matemática e Informática; Matematika eta Informatika Ingeniaritza; Institute of Smart Cities - ISC
In this paper we deal with solving efficiently 2D linear parabolic singularly perturbed systems of convection–diffusion type. We analyze only the case of a system of two equations where both of them feature the same diffusion parameter. Nevertheless, the method is easily extended to systems with an arbitrary number of equations which have the same diffusion coefficient. The fully discrete numerical method combines the upwind finite difference scheme, to discretize in space, and the fractional implicit Euler method, together with a splitting by directions and components of the reaction–convection–diffusion operator, to discretize in time. Then, if the spatial discretization is defined on an appropriate piecewise uniform Shishkin type mesh, the method is uniformly convergent and it is first order in time and almost first order in space. The use of a fractional step method in combination with the splitting technique to discretize in time, means that only tridiagonal linear systems must be solved at each time level of the discretization. Moreover, we study the order reduction phenomenon associated with the time dependent boundary conditions and we provide a simple way of avoiding it. Some numerical results, which corroborate the theoretical established properties of the method, are shown.
Open Access
A linearly implicit splitting method for solving time dependent semilinear reaction-diffusion systems
(Springer, 2020) Clavero, Carmelo; Jorge Ulecia, Juan Carlos; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC
In this paper we deal with the efficient resolution of a coupled system of two one dimensional, time dependent, semilinear parabolic singularly perturbed partial differential equations of reaction-diffusion type, with distinct diffusion parameters which may have different orders of magnitude. The numerical method is based on a linearized version of the fractional implicit Euler method, which avoids the use of iterative methods, and a splitting by components to discretize in time; so, only tridiagonal linear systems are involved in the time integration process. Consequently, the computational cost of the proposed method is lower than classical schemes used for the same type of problems. The solution of this singularly perturbed problem features layers, what are resolved on an appropriate piecewise uniform mesh of Shishkin type. We show that the method is uniformly convergent of first order in time and of almost second order in space. Numerical results are presented to corroborate the theoretical results.
Open Access
Solving efficiently one dimensional parabolic singularly perturbed reaction¿diffusion systems: a splitting by components
(Elsevier, 2018) Clavero, Carmelo; Jorge Ulecia, Juan Carlos; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC
In this paper we consider 1D parabolic singularly perturbed systems of reaction¿diffusion type which are coupled in the reaction term. The numerical scheme, used to approximate the exact solution, combines the fractional implicit Euler method and a splitting by components to discretize in time, and the classical central finite differences scheme to discretize in space. The use of the fractional Euler method combined with the splitting by components means that only tridiagonal linear systems must be solved to obtain the numerical solution. For simplicity, the analysis is presented in a complete form only in the case of systems which have two equations, but it can be easily extended to an arbitrary number of equations. If a special nonuniform mesh in space is used, the method is uniformly and unconditionally convergent, having first order in time and almost second order in space. Some numerical results are shown which corroborate in practice the theoretical ones.
Open Access
Embedded pairs of fractional step Runge-Kutta methods and improved domain decomposition techniques for parabolic problems
(Springer, 2007) Portero Egea, Laura; Jorge Ulecia, Juan Carlos; Ingeniería Matemática e Informática; Matematika eta Informatika Ingeniaritza
In this paper we design and apply new embedded pairs of Frac- tional Step Runge-Kutta methods to the e±cient solution of multidimensional parabolic problems. These time integrators are combined with a suitable split- ting of the elliptic operator subordinated to a decomposition of the spatial domain and a standard spatial discretization. With this technique we ob- tain parallel algorithms which have the main advantages of classical domain decomposition methods and, besides, avoid iterative processes like Schwarz iterations, typical of them. The use of these embedded methods permits a fast variable step time integration process.
Open Access
An efficient and uniformly convergent scheme for one-dimensional parabolic singularly perturbed semilinear systems of reaction-diffusion type
(Springer, 2020) Clavero, Carmelo; Jorge Ulecia, Juan Carlos; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC
Open Access
Parallel solution of nonlinear parabolic problems on logically rectangular grids
(Springer, 2007) Arrarás Ventura, Andrés; Portero Egea, Laura; Jorge Ulecia, Juan Carlos; Ingeniería Matemática e Informática; Matematika eta Informatika Ingeniaritza
This work deals with the efficient numerical solution of nonlinear transient flow problems posed on two-dimensional porous media of general geometry. We first consider a spatial semidiscretization of such problems by using a cell-centered finite difference scheme on a logically rectangular grid. The resulting nonlinear stiff initial-value problems are then integrated in time by means of a fractional step method, combined with a decomposition of the flow domain into a set of overlapping subdomains and a linearization procedure which involves suitable Taylor expansions. The proposed algorithm reduces the original problem to the solution of several linear systems per time step. Moreover, each one of such systems can be directly decomposed into a set of uncoupled linear subsystems which can be solved in parallel. A numerical example illustrates the unconditionally convergent behaviour of the method in the last section of the paper.
Embargo
New fractional step Runge-Kutta-Nyström methods up to order three
(Elsevier, 2020) Bujanda Cirauqui, Blanca; Moreta, M. Jesús; Jorge Ulecia, Juan Carlos; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas
Fractional Step Runge–Kutta–Nyströ (FSRKN) methods have been revealed to be an excellent option to integrate numerically many multidimensional evolution models governed by second order in time partial differential equations. These methods, combined with suitable spatial discretizations, lead to strong computational cost reductions respect to many classical implicit time integrators. In this paper, we present the construction process of several implicit FSRKN methods of two and three levels which attain orders up to three and satisfy adequate stability properties. We have also performed some numerical experiments in order to show the unconditionally convergent behavior of these schemes as well as their computational advantages.