1,759 research outputs found
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
An almost symmetric Strang splitting scheme for the construction of high order composition methods
In this paper we consider splitting methods for nonlinear ordinary
differential equations in which one of the (partial) flows that results from
the splitting procedure can not be computed exactly. Instead, we insert a
well-chosen state into the corresponding nonlinearity ,
which results in a linear term whose exact flow can be
determined efficiently. Therefore, in the spirit of splitting methods, it is
still possible for the numerical simulation to satisfy certain properties of
the exact flow. However, Strang splitting is no longer symmetric (even though
it is still a second order method) and thus high order composition methods are
not easily attainable. We will show that an iterated Strang splitting scheme
can be constructed which yields a method that is symmetric up to a given order.
This method can then be used to attain high order composition schemes. We will
illustrate our theoretical results, up to order six, by conducting numerical
experiments for a charged particle in an inhomogeneous electric field, a
post-Newtonian computation in celestial mechanics, and a nonlinear population
model and show that the methods constructed yield superior efficiency as
compared to Strang splitting. For the first example we also perform a
comparison with the standard fourth order Runge--Kutta methods and find
significant gains in efficiency as well better conservation properties
Solving linear and non-linear stiff system of ordinary differential equations by multistage adomian decomposition method
In this paper, linear and non-linear stiff systems of ordinary differential equations are solved by the classical Adomian decomposition method (ADM) and the multistage Adomian decomposition method (MADM). The MADM is a technique adapted from the standard Adomian decomposition method (ADM) where standard ADM is converted into a hybrid numeric-analytic method called the multistage ADM (MADM). The MADM is tested for several examples. Comparisons with an explicit Runge-Kutta-type method (RK) and the classical ADM demonstrate the limitations of ADM and promising capability of the MADM for solving stiff initial value problems (IVPs)
Extrapolation-based implicit-explicit general linear methods
For many systems of differential equations modeling problems in science and
engineering, there are natural splittings of the right hand side into two
parts, one non-stiff or mildly stiff, and the other one stiff. For such systems
implicit-explicit (IMEX) integration combines an explicit scheme for the
non-stiff part with an implicit scheme for the stiff part.
In a recent series of papers two of the authors (Sandu and Zhang) have
developed IMEX GLMs, a family of implicit-explicit schemes based on general
linear methods. It has been shown that, due to their high stage order, IMEX
GLMs require no additional coupling order conditions, and are not marred by
order reduction.
This work develops a new extrapolation-based approach to construct practical
IMEX GLM pairs of high order. We look for methods with large absolute stability
region, assuming that the implicit part of the method is A- or L-stable. We
provide examples of IMEX GLMs with optimal stability properties. Their
application to a two dimensional test problem confirms the theoretical
findings
Implementation of a new third order weighted Runge-Kutta formula based on Centroidal Mean for solving stiff initial value problems
A new third order weighted Runge-Kutta formula based on Centroidal Mean(CeM) is derived and implemented. To illustrate the effectiveness of the method, a stiff problem has been chosen and compared with the classical fourth order Runge-Kutta method and the third order weighted Runge-Kutta method based on Contraharmonic Mean(CoH). The stability of the new method is analysed. The investigation undertaken in the study reveals that the third order RK method based on CeM suits very well and indicates that this method is superior compared to the other methods discussed for the stiff initial value problems
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