4,519 research outputs found
Galerkin/Runge-Kutta discretizations for parabolic equations with time dependent coefficients
A new class of fully discrete Galerkin/Runge-Kutta methods is constructed and analyzed for linear parabolic initial boundary value problems with time dependent coefficients. Unlike any classical counterpart, this class offers arbitrarily high order convergence while significantly avoiding what has been called order reduction. In support of this claim, error estimates are proved, and computational results are presented. Additionally, since the time stepping equations involve coefficient matrices changing at each time step, a preconditioned iterative technique is used to solve the linear systems only approximately. Nevertheless, the resulting algorithm is shown to preserve the original convergence rate while using only the order of work required by the base scheme applied to a linear parabolic problem with time independent coefficients. Furthermore, it is noted that special Runge-Kutta methods allow computations to be performed in parallel so that the final execution time can be reduced to that of a low order method
Spatial Manifestations of Order Reduction in Runge-Kutta Methods for Initial Boundary Value Problems
This paper studies the spatial manifestations of order reduction that occur
when time-stepping initial-boundary-value problems (IBVPs) with high-order
Runge-Kutta methods. For such IBVPs, geometric structures arise that do not
have an analog in ODE IVPs: boundary layers appear, induced by a mismatch
between the approximation error in the interior and at the boundaries. To
understand those boundary layers, an analysis of the modes of the numerical
scheme is conducted, which explains under which circumstances boundary layers
persist over many time steps. Based on this, two remedies to order reduction
are studied: first, a new condition on the Butcher tableau, called weak stage
order, that is compatible with diagonally implicit Runge-Kutta schemes; and
second, the impact of modified boundary conditions on the boundary layer theory
is analyzed.Comment: 41 pages, 9 figure
Optimized explicit Runge-Kutta schemes for the spectral difference method applied to wave propagation problems
Explicit Runge-Kutta schemes with large stable step sizes are developed for
integration of high order spectral difference spatial discretization on
quadrilateral grids. The new schemes permit an effective time step that is
substantially larger than the maximum admissible time step of standard explicit
Runge-Kutta schemes available in literature. Furthermore, they have a small
principal error norm and admit a low-storage implementation. The advantages of
the new schemes are demonstrated through application to the Euler equations and
the linearized Euler equations.Comment: 37 pages, 3 pages of appendi
Comparison of efficiency among different techniques to avoid order reduction with Strang splitting
In this paper, we offer a comparison in terms of computational efficiency between two techniques to avoid order reduction when using Strang method to integrate nonlinear initial boundary value problems with time-dependent boundary conditions. We see that it is important to consider an exponential method for the integration of the linear nonhomogeneous and stiff part in the technique by Einkemmer et al. so that the latter is comparable in efficiency with that suggested by Alonso et al.
Some other advantages of the technique suggested by Alonso et al. are stated in the conclusions
How to avoid order reduction when Lawson methods integrate nonlinear initial boundary value problems
It is well known that Lawson methods suffer from a severe order reduction when integrating initial boundary value problems where the solutions are not periodic in space or do not satisfy enough conditions of annihilation on the boundary. However, in a previous paper, a modification of Lawson quadrature rules has been suggested so that no order reduction turns up when integrating linear problems subject to time-dependent boundary conditions. In this paper, we describe and thoroughly analyse a technique to avoid also order reduction when integrating nonlinear problems. This is very useful because, given any Runge–Kutta method of any classical order, a Lawson method can be constructed associated to it for which the order is conserved.This work was funded by Ministerio de Ciencia e Innovación and Regional Development European Funds through project PGC2018-101443-B-I00 and by Junta de Castilla y León and Feder through projects VA169P20
Avoiding order reduction when integrating linear initial boundary value problems with Lawson methods
Exponential Lawson methods are well known to have a severe order reduction when integrating stiff
problems. In a previous article, the precise order observed with Lawson methods when integrating linear
problems is justified in terms of different conditions of annihilation on the boundary. In fact, the analysis
of convergence with all exponential methods when applied to parabolic problems has always been performed
under assumptions of vanishing boundary conditions for the solution. In this article, we offer a
generalization of Lawson methods to approximate problems with nonvanishing and even time-dependent
boundary values. This technique is cheap and allows to avoid completely order reduction independently
of having vanishing or nonvanishing boundary conditions.Este trabajo forma parte del proyecto de investigación: MTM 2015-66837-
Avoiding order reduction phenomenon for general linear methods when integrating linear problems with time dependent boundary values
Producción CientíficaWhen applied to stiff problems, the effective order of convergence of general linear methods is governed by their stage order, which is less than or equal to the classical order of the method. This produces an order reduction phenomenon, present in all general linear methods except those with high stage order, in a manner similar to that observed in other time integrators with internal stages.
In this paper, we investigate the order reduction which arises when general linear methods are used as time integrators when using the method of lines for solving numerically initial boundary value problems with time dependent boundary values.
We propose a technique, based on making an appropriate choice of the boundary values for the internal stages, with which it is possible to recover one unit of order, as we prove in this work. As expected, this implies a considerable improvement for the general linear methods suffering order reduction. Moreover, numerical experiments show that the improvement is not only in these cases, but that, even when the order reduction is not expected, the size of the errors is drastically reduced by using the technique proposed in this paper.Ministerio de Ciencia e Innovación y Ministerio de Universidades (project PGC2018-101443-B-100)Junta de Castilla y León (Grant numbers VA169P20 and VA193P20
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