579 research outputs found
Strong and auxiliary forms of the semi-Lagrangian method for incompressible flows
We present a review of the semi-Lagrangian method for advection-diusion and incompressible Navier-Stokes equations discretized with high-order methods. In particular, we compare the strong form where the departure points are computed directly via backwards integration with the auxiliary form where an auxiliary advection equation is solved instead; the latter is also referred to as Operator Integration Factor Splitting (OIFS) scheme. For intermediate size of time steps the auxiliary form is preferrable but for large time steps only the strong form is stable
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
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
DIRK Schemes with High Weak Stage Order
Runge-Kutta time-stepping methods in general suffer from order reduction: the
observed order of convergence may be less than the formal order when applied to
certain stiff problems. Order reduction can be avoided by using methods with
high stage order. However, diagonally-implicit Runge-Kutta (DIRK) schemes are
limited to low stage order. In this paper we explore a weak stage order
criterion, which for initial boundary value problems also serves to avoid order
reduction, and which is compatible with a DIRK structure. We provide specific
DIRK schemes of weak stage order up to 3, and demonstrate their performance in
various examples.Comment: 10 pages, 5 figure
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Space-time Methods for Time-dependent Partial Differential Equations
Modern discretizations of time-dependent PDEs consider the full problem in the space-time cylinder and aim to overcome limitations of classical approaches such as the method of lines (first discretize in space and then solve the resulting ODE) and the Rothe method (first discretize in time and then solve the PDE). A main advantage of a holistic space-time method is the direct access to space-time adaptivity and to the backward problem (required for the dual problem in optimization or error control). Moreover, this allows for parallel solution strategies simultaneously in time and space.
Several space-time concepts where proposed (different conforming and nonconforming space-time finite elements, the parareal method, wavefront relaxation etc.) but this topic has become a rapidly growing field in numerical analysis and scientific computing. In this workshop the focus is the development of adaptive and flexible space-time discretization methods for solving parabolic and hyperbolic space-time partial differential equations
A-stable Runge-Kutta methods for semilinear evolution equations
We consider semilinear evolution equations for which the linear part
generates a strongly continuous semigroup and the nonlinear part is
sufficiently smooth on a scale of Hilbert spaces. In this setting, we prove the
existence of solutions which are temporally smooth in the norm of the lowest
rung of the scale for an open set of initial data on the highest rung of the
scale. Under the same assumptions, we prove that a class of implicit,
-stable Runge--Kutta semidiscretizations in time of such equations are
smooth as maps from open subsets of the highest rung into the lowest rung of
the scale. Under the additional assumption that the linear part of the
evolution equation is normal or sectorial, we prove full order convergence of
the semidiscretization in time for initial data on open sets. Our results
apply, in particular, to the semilinear wave equation and to the nonlinear
Schr\"odinger equation
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