Blanca Bujanda

A high order uniformly convergent alternating direction scheme for time dependent reaction–diffusion singularly perturbed problems

In this work we design and analyze an efficient numerical method to solve two dimensional initial-boundary value reaction–diffusion problems, for which the diffusion parameter can be very small with respect to the reaction term. The method is defined by combining the Peaceman and Rachford alternating direction method to discretize in time, together with a HODIE finite difference scheme constructed on a tailored mesh. We prove that the resulting scheme is ε-uniformly convergent of second order in time and of third order in spatial variables. Some numerical examples illustrate the efficiency of the method and the orders of uniform convergence proved theoretically. We also show that it is easy to avoid the well-known order reduction phenomenon, which is usually produced in the time integration process when the boundary conditions are time dependent.

A Combined Fractional Step Domain Decomposition Method for the Numerical Integration of Parabolic Problems

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.

High Order Uniformly Convergent Fractional Step RK Methods and HODIE Finite Difference Schemes for 2D Evolutionary Convection-Diffusion Problems

In this work we present a numerical method to solve linear time dependent two dimensional singularly perturbed problems of convection-diffusion type with dominat- ing convection term; this class of problems is characterized by the presence of a regular boundary layer in the output boundary of the spatial domain. The method combines the alternating direction technique, based on an A-stable third order RK method, with a third order HODIE finite difference scheme of classical type, i.e., exact only on polynomial functions, constructed on a special spatial mesh of Shishkin type. We show that, under appropriate restrictions between the discretization parameters, the method is uniformly convergent with respect to the diffusion parameter, having order three (except by a loga- rithmic factor) in the maximum norm. The method provides the computational advantages of the splitting technique and also the efficiency provided by high order methods, achieving good approximations of the solution in the whole domain, including the boundary layer region. We show some numerical results validating in practice the good properties of the method.

Convergent expansions of the incomplete gamma functions in terms of elementary functions

We consider the incomplete gamma function γ(a,z) for ℜa > 0 and z ∈ ℂ. We derive several convergent expansions of z−aγ(a,z) in terms of exponentials and rational functions of z that hold uniformly in z with ℜz bounded from below. These expansions, multiplied by ez, are expansions of ezz−aγ(a,z) uniformly convergent in z with ℜz bounded from above. The expansions are accompanied by realistic error bounds.

Avoiding the order reduction when solving second-order in time PDEs with Fractional Step Runge-Kutta-Nyström methods

We study some of the main features of Fractional Step Runge-Kutta-Nystrom 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.

A Third Order Linearly Implicit Fractional Step Method for Semilinear Parabolic Problems

In this paper a new efficient linearly implicit time integrator for semilinear multidimensional parabolic problems is proposed. This method preserves the advantages, in terms of computational cost reduction, of the classical fractional step methods for linear parabolic problems. We show some numerical tests for illustrating that this method combined with standard space discretization techniques, provides efficient numerical algorithms capable of computing stable numerical solutions without restrictions between the time step and the mesh size.

Convergent expansions of the confluent hypergeometric functions in terms of elementary functions

This research was supported by the Spanish Ministry of Economia y Competitividad, projects MTM2014-53178-P and TEC2013-45585-C2-1-R. The Universidad Publica de Navarra is acknowledged for its financial support.

Fractional Step Runge-Kutta Methods for the Resolution of Two Dimensional Time Dependent Coefficient Convection-Diffusion Problems

In this paper we obtain a unconditional convergence result for discretization methods of type Fractional Steps Runge-Kutta, which are highly efficient in the numerical resolution of parabolic problems whose coefficients depend on time. These methods combined with standard spatial discretizations will provide totally discrete algorithms with low computational cost and high order of accuracy in time. We will show the efficiency of such methods, in combination with upwind difference schemes on special meshes, to integrate numerically singularly perturbed evolutionary convection-diffusion problems.