The Experts below are selected from a list of 14043 Experts worldwide ranked by ideXlab platform
J. C. Jorge - One of the best experts on this subject based on the ideXlab platform.
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Variable Step-size Fractional Step Runge-Kutta methods for time-dependent partial differential equations
Applied Numerical Mathematics, 2012Co-Authors: Laura Portero, Andrés Arrarás, J. C. JorgeAbstract:Fractional Step Runge-Kutta methods are a class of additive Runge-Kutta schemes that provide efficient time discretizations for evolutionary partial differential equations. This efficiency is due to appropriate decompositions of the elliptic operator involving the spatial derivatives. In this work, we tackle the design and analysis of embedded pairs of Fractional Step Runge-Kutta methods. Such methods suitably estimate the local error at each time Step, thus providing efficient variable Step-size time integrations. Finally, some numerical experiments illustrate the behaviour of the proposed algorithms.
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Order conditions for linearly implicit Fractional Step Runge–Kutta methods
IMA Journal of Numerical Analysis, 2007Co-Authors: B. Bujanda, J. C. JorgeAbstract:In this paper, we study the consistency of a variant of Fractional Step Runge-Kutta methods. These methods are designed to integrate efficiently semi-linear multidimensional parabolic problems by means of linearly implicit time integration processes. Such time discretization procedures are also related to a splitting of the space differential operator (or the spatial discretization of it) as a sum of ‘simpler’ linear differential operators and a nonlinear term.
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A generalization of Peaceman–Rachford Fractional Step method
Journal of Computational and Applied Mathematics, 2006Co-Authors: L. Portero, J. C. JorgeAbstract:AbstractIn this paper we develop a set of time integrators of type Fractional Step Runge–Kutta methods which generalize the time integrator involved in the classical Peaceman–Rachford scheme. Combining a time semidiscretization of this type with a standard spatial discretization, we obtain a totally discrete algorithm capable of discretizing efficiently a general parabolic problem if suitable splittings of the elliptic operator are considered. We prove that our proposal is second order consistent and stable even for an operator splitting in m terms which do not necessarily commute. Finally, we illustrate the theoretical results with various applications such as alternating directions or evolutionary domain decomposition
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A generalization of Peaceman-Rachford Fractional Step method
Journal of Computational and Applied Mathematics, 2005Co-Authors: Laura Portero, J. C. JorgeAbstract:In this paper we develop a set of time integrators of type Fractional Step Runge-Kutta methods which generalize the time integrator involved in the classical Peaceman-Rachford scheme. Combining a time semidiscretization of this type with a standard spatial discretization, we obtain a totally discrete algorithm capable of discretizing efficiently a general parabolic problem if suitable splittings of the elliptic operator are considered. We prove that our proposal is second order consistent and stable even for an operator splitting in m terms which do not necessarily commute. Finally, we illustrate the theoretical results with various applications such as alternating directions or evolutionary domain decomposition.
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Spectral-Fractional Step Runge–Kutta discretizations for initial boundary value problems with time dependent boundary conditions
Mathematics of Computation, 2004Co-Authors: Isaías Alonso-mallo, B. Cano, J. C. JorgeAbstract: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.
Ramon Codina - One of the best experts on this subject based on the ideXlab platform.
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On the Design of Algebraic Fractional Step Methods for Viscoelastic Incompressible Flows
SEMA SIMAI Springer Series, 2018Co-Authors: Ramon CodinaAbstract:Classical Fractional Step methods for viscous incompressible flows aim to uncouple the calculation of the velocity and the pressure. In the case of viscoelastic flows, a new variable appears, namely, a stress, which has an elastic and a viscous contribution. The purpose of this article is to present two families of Fractional Step methods for the time integration of this type of flows whose objective is to permit the uncoupled calculation of velocities, stresses and pressure, both families designed at the algebraic level. This means that the splitting of the equations is introduced once the spatial and the temporal discretizations have been performed. The first family is based on the extrapolation of the pressure and the stress in order to predict a velocity, then the calculation of a new stress, the pressure and then a correction to render the scheme stable. The second family has a discrete pressure Poisson equation as starting point; in this equation, velocities and stresses are extrapolated to compute a pressure, and from this pressure stresses and velocities can then be computed. This work presents an overview of methods previously proposed in our group, as well as some new schemes in the case of the second family.
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First, second and third order Fractional Step methods for the three-field viscoelastic flow problem
Journal of Computational Physics, 2015Co-Authors: Ernesto Castillo, Ramon CodinaAbstract:In this paper, three different Fractional Step methods are designed for the three-field viscoelastic flow problem, whose variables are velocity, pressure and elastic stress. The starting point of our methods is the same as for classical pressure segregation algorithms used in the Newtonian incompressible Navier-Stokes problem. These methods can be understood as an inexact LU block factorization of the original system matrix of the fully discrete problem and are designed at the pure algebraic level. The final schemes allow one to solve the problem in a fully decoupled form, where each equation (for velocity, pressure and elastic stress) is solved separately. The first order scheme is obtained from a straightforward segregation of pressure and elastic stress in the momentum equation, whereas the key point for the second order scheme is a first order extrapolation of these variables. The third order Fractional Step method relies on Yosida's scheme. Referring to the spatial discretization, either the Galerkin method or a stabilized finite element formulation can be used. We describe the Fractional Step methods first assuming the former, and then we explain the modifications introduced by the stabilized formulation we employ and that has been proposed in a previous work. This discretization in space shows very good stability, permitting in particular the use of equal interpolation for all variables.
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Pressure Stability in Fractional Step Finite Element Methods for Incompressible Flows
Journal of Computational Physics, 2001Co-Authors: Ramon CodinaAbstract:Abstract The objective of this paper is to analyze the pressure stability of Fractional Step finite element methods for incompressible flows that use a pressure Poisson equation. For the classical first-order projection method, it is shown that there is a pressure control which depends on the time Step size, and therefore there is a lower bound for this time Step for stability reasons. The situation is much worse for a second-order scheme in which part of the pressure gradient is kept in the momentum equation. The pressure stability in this case is extremely weak. To overcome these shortcomings, a stabilized Fractional Step finite element method is also considered, and its stability is analyzed. Some simple numerical examples are presented to support the theoretical results.
Paola Gervasio - One of the best experts on this subject based on the ideXlab platform.
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Algebraic Fractional-Step Schemes for Time-Dependent Incompressible Navier–Stokes Equations
Journal of Scientific Computing, 2006Co-Authors: Paola Gervasio, Fausto SaleriAbstract:The numerical investigation of a recent family of algebraic Fractional-Step methods (the so called Yosida methods) for the solution of the incompressible time-dependent Navier–Stokes equations is presented. A comparison with the Karniadakis–Israeli–Orszag method Karniadakis et al. (1991, J. Comput. Phys. 97, 414–443) is carried out. The high accuracy in time of these schemes well combines with the high accuracy in space of spectral methods
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Algebraic Fractional-Step Schemes for Time-Dependent Incompressible Navier---Stokes Equations
Journal of Scientific Computing, 2006Co-Authors: Paola Gervasio, Fausto SaleriAbstract:The numerical investigation of a recent family of algebraic Fractional-Step methods (the so called Yosida methods) for the solution of the incompressible time-dependent Navier---Stokes equations is presented. A comparison with the Karniadakis---Israeli---Orszag method Karniadakis et al. (1991, J. Comput. Phys. 97, 414---443) is carried out. The high accuracy in time of these schemes well combines with the high accuracy in space of spectral methods.
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Fractional Step METHODS FOR SPECTRAL APPROXIMATION OF ADVECTION-DIFFUSION EQUATIONS
Mathematical Models and Methods in Applied Sciences, 1996Co-Authors: Paola GervasioAbstract:Several classical Fractional Step schemes are proposed for the spectral approximation of advection-diffusion equation in two-dimensional geometries. Suitable boundary conditions are studied in order to preserve the accuracy of the schemes at each Step. An example is given for the application of the Fractional Step schemes to solve problems with large Peclet number.
P. Kjellgren - One of the best experts on this subject based on the ideXlab platform.
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A semi-implicit Fractional Step finite element method for viscous incompressible flows
Computational Mechanics, 1997Co-Authors: P. KjellgrenAbstract:This paper describes a new semi-implicit finite element algorithm for time-dependent viscous incompressible flows. The algorithm is of a general type and can handle both low and high Reynolds number flows, although the emphasis is on convection dominated flows. An explicit three-Step method is used for the convection term and an implicit trapezoid method for the diffusion term. The consistent mass matrix is only used in the momentum phase of the Fractional Step algorithm while the lumped mass matrix is used in the pressure phase and in the pressure Poisson equation. An accuracy and stability analysis of the algorithm is provided for the pure convection equation. Two different types of boundary conditions for the end-of-Step velocity of the Fractional Step algorithm have been investigated.
Young-bae Kim - One of the best experts on this subject based on the ideXlab platform.
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canonical Fractional Step methods and consistent boundary conditions for the incompressible navier stokes equations
Journal of Computational Physics, 2001Co-Authors: Moon Joo Lee, Young-bae KimAbstract:Abstract An account of second-order Fractional-Step methods and boundary conditions for the incompressible Navier–Stokes equations is presented. The goals of the work were (i) identification and analysis of all possible splitting methods of second-order splitting accuracy, and (ii) determination of consistent boundary conditions that yield second-order-accurate solutions. Exact and approximate block-factorization techniques were used to construct second-order splitting methods. It has been found that only three canonical types (D, P, and M) of splitting methods are nondegenerate, and all other second-order splitting schemes are either degenerate or equivalent to them. Investigation of the properties of the canonical methods indicates that a method of type D is recommended for computations in which zero divergence is preferred, while a method of type P is better suited to cases where highly accurate pressure is more desirable. The consistent boundary conditions on the tentative velocity and pressure have been determined by a procedure that consists of approximation of the split equations and the boundary limit of the result. It has been found that the pressure boundary condition is independent of the type of Fractional-Step methods. The consistent boundary conditions on the tentative velocity were determined in terms of the natural boundary condition and derivatives of quantities available at the current time Step (to be evaluated by extrapolation). Second-order Fractional-Step methods that admit the zero-pressure-gradient boundary condition have been derived by using a transformation that involves the “delta form” pressure. The boundary condition on the new tentative velocity becomes greatly simplified due to improved accuracy built into the transformation.
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Canonical Fractional-Step methods and consistent boundary conditions for the incompressible Navier—Stokes equations
Journal of Computational Physics, 2001Co-Authors: Moon Joo Lee, Young-bae KimAbstract:Abstract An account of second-order Fractional-Step methods and boundary conditions for the incompressible Navier–Stokes equations is presented. The goals of the work were (i) identification and analysis of all possible splitting methods of second-order splitting accuracy, and (ii) determination of consistent boundary conditions that yield second-order-accurate solutions. Exact and approximate block-factorization techniques were used to construct second-order splitting methods. It has been found that only three canonical types (D, P, and M) of splitting methods are nondegenerate, and all other second-order splitting schemes are either degenerate or equivalent to them. Investigation of the properties of the canonical methods indicates that a method of type D is recommended for computations in which zero divergence is preferred, while a method of type P is better suited to cases where highly accurate pressure is more desirable. The consistent boundary conditions on the tentative velocity and pressure have been determined by a procedure that consists of approximation of the split equations and the boundary limit of the result. It has been found that the pressure boundary condition is independent of the type of Fractional-Step methods. The consistent boundary conditions on the tentative velocity were determined in terms of the natural boundary condition and derivatives of quantities available at the current time Step (to be evaluated by extrapolation). Second-order Fractional-Step methods that admit the zero-pressure-gradient boundary condition have been derived by using a transformation that involves the “delta form” pressure. The boundary condition on the new tentative velocity becomes greatly simplified due to improved accuracy built into the transformation.
