11,174 research outputs found
High order explicit exponential Runge-Kutta methods for the weak approximation of solutions of stochastic differential equations
We are concerned with numerical methods which give weak approximations for stiff Ito stochastic differential equations (SDEs). It is well known that the numerical solution of stiff SDEs leads to a stepsize reduction when explicit methods are used. However, there are some classes of explicit methods that are well suited to solving some types of stiff SDEs. One such class is the class of stochastic orthogonal Runge-Kutta Chebyshev (SROCK) methods. SROCK methods reduce to Runge-Kutta Chebyshev methods when applied to ordinary differential equations (ODEs). Another promising class of methods is the class of explicit methods that reduce to explicit exponential Runge-Kutta (RK) methods when applied to semilinear ODEs. In this paper, we will propose new exponential RK methods which achieve weak order one or two for multi-dimensional, non-commutative SDEs with a semilinear drift term, whereas they are of order one, two or three for semilinear ODEs. We will analytically investigate their stability properties in mean square, and will check their performance in numerical examples
Computer solution of non-linear integration formula for solving initial value problems
This thesis is concerned with the numerical
solutions of initial value problems with ordinary
differential equations and covers
single step integration methods.
focus is to study the numerical
the various aspects of
Specifically, its main
methods of non-linear
integration formula with a variety of means based on the
Contraharmonic mean (C˳M) (Evans and Yaakub [1995]), the
Centroidal mean (C˳M) (Yaakub and Evans [1995]) and the
Root-Mean-Square (RMS) (Yaakub and Evans [1993]) for
solving initial value problems.
the applications of the second
It includes a study of
order C˳M method for
parallel implementation of extrapolation methods for
ordinary differential equations with the ExDaTa schedule
by Bahoshy [1992]. Another important topic presented in
this thesis is that a fifth order five-stage explicit
Runge Kutta method or weighted Runge Kutta formula [Evans
and Yaakub [1996]) exists which is contrary to Butcher
[1987] and the theorem in Lambert ([1991] ,pp 181).
The thesis is organized as follows. An introduction
to initial value problems in ordinary differential
equations and parallel computers and software in Chapter
1, the basic preliminaries and fundamental concepts in
mathematics, an algebraic manipulation package, e.g.,
Mathematica and basic parallel processing techniques are
discussed in Chapter 2. Following in Chapter 3 is a
survey of single step methods to solve ordinary
differential equations. In this chapter, several single
step methods including the Taylor series method, Runge
Kutta method and a linear multistep method for non-stiff
and stiff problems are also considered.
Chapter 4 gives a new Runge Kutta formula for
solving initial value problems using the Contraharmonic
mean (C˳M), the Centroidal mean (C˳M) and the Root-MeanSquare
(RMS). An error and stability analysis for these
variety of means and numerical examples are also
presented. Chapter 5 discusses the parallel
implementation on the Sequent 8000 parallel computer of
the Runge-Kutta contraharmonic mean (C˳M) method with
extrapolation procedures using explicit
assignment scheduling
Kutta RK(4, 4) method
(EXDATA) strategies. A
is introduced and the
data task
new Rungetheory
and
analysis of its properties are investigated and compared
with the more popular RKF(4,5) method, are given in
Chapter 6. Chapter 7 presents a new integration method
with error control for the solution of a special class of
second order ODEs. In Chapter 8, a new weighted Runge-Kutta
fifth order method with 5 stages is introduced. By
comparison with the currently recommended RK4 ( 5) Merson
and RK5(6) Nystrom methods, the new method gives improved
results. Chapter 9 proposes a new fifth order Runge-Kutta
type method for solving oscillatory problems by the use
of trigonometric polynomial interpolation which extends
the earlier work of Gautschi [1961]. An analysis of the
convergence and stability of the new method is given with
comparison with the standard Runge-Kutta methods.
Finally, Chapter 10 summarises and presents
conclusions on the topics
discussed throughout the thesis
Rational Approximation Method for Stiff Initial Value Problems
While purely numerical methods for solving ordinary differential equations (ODE), e.g., Runge–Kutta methods, are easy to implement, solvers that utilize analytical derivations of the right-hand side of the ODE, such as the Taylor series method, outperform them in many cases. Nevertheless, the Taylor series method is not well-suited for stiff problems since it is explicit and not A-stable. In our paper, we present a numerical-analytical method based on the rational approximation of the ODE solution, which is naturally A- and A(α)-stable. We describe the rational approximation method and consider issues of order, stability, and adaptive step control. Finally, through examples, we prove the superior performance of the rational approximation method when solving highly stiff problems, comparing it with the Taylor series and Runge–Kutta methods of the same accuracy order
Multi-Adaptive Time-Integration
Time integration of ODEs or time-dependent PDEs with required resolution of
the fastest time scales of the system, can be very costly if the system
exhibits multiple time scales of different magnitudes. If the different time
scales are localised to different components, corresponding to localisation in
space for a PDE, efficient time integration thus requires that we use different
time steps for different components.
We present an overview of the multi-adaptive Galerkin methods mcG(q) and
mdG(q) recently introduced in a series of papers by the author. In these
methods, the time step sequence is selected individually and adaptively for
each component, based on an a posteriori error estimate of the global error.
The multi-adaptive methods require the solution of large systems of nonlinear
algebraic equations which are solved using explicit-type iterative solvers
(fixed point iteration). If the system is stiff, these iterations may fail to
converge, corresponding to the well-known fact that standard explicit methods
are inefficient for stiff systems. To resolve this problem, we present an
adaptive strategy for explicit time integration of stiff ODEs, in which the
explicit method is adaptively stabilised by a small number of small,
stabilising time steps
A class of implicit-explicit two-step Runge-Kutta methods
This work develops implicit-explicit time integrators based on two-step Runge-Kutta methods.
The class of schemes of interest is characterized by linear invariant
preservation and high stage orders. Theoretical consistency and stability analyses are performed to reveal the properties of these methods. The new framework offers extreme flexibility
in the construction of partitioned integrators, since no coupling conditions are necessary.
Moreover, the methods are not plagued by severe order reduction, due to their high stage orders.
Two practical schemes of orders four and six are constructed, and are used to solve several test problems.
Numerical results confirm the theoretical findings
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
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