36,580 research outputs found
Spatially partitioned embedded Runge-Kutta Methods
We study spatially partitioned embedded Runge–Kutta (SPERK) schemes for partial differential equations (PDEs), in which each of the component schemes is applied over a different part of the spatial domain. Such methods may be convenient for problems in which the smoothness of the solution or the magnitudes of the PDE coefficients vary strongly in space. We focus on embedded partitioned methods as they offer greater efficiency and avoid the order reduction that may occur in non-embedded schemes. We demonstrate that the lack of conservation in partitioned schemes can lead to non-physical effects and propose conservative additive schemes based on partitioning the fluxes rather than the ordinary differential equations. A variety of SPERK schemes are presented, including an embedded pair suitable for the time evolution of fifth-order weighted non-oscillatory (WENO) spatial discretizations. Numerical experiments are provided to support the theory
Runge-Kutta Methods for Solving Ordinary and Delay Differential Equations
An introduction to Runge-Kutta methods for the solution of ordinary differential equations (ODEs) is introduced. The technique of using Singly Diagonally Implicit Runge-Kutta (SDIRK) method for the integration of stiff and non-stiff ODEs has been widely accepted, this is because SDIRK method is computationally efficient and stiffly stable. Consequently embedded SDIRK method of fourth-order six stage in fifth-order seven stage which has the property that the first row of the coefficient matrix is equal to zero and the last row of the coefficient matrix is equal to the vector output value is constructed. The stability region of the method when applied to linear ODE is given. Numerical results when stiff and non-stiff first order ODEs are solved using the method are tabulated and compared with the method in current use.
Introduction to delay differential equations (DDEs) and the areas where they arise are given. A brief discussion on Runge-Kutta method when adapted to delay differential equation is introduced. SDIRK method which has been derived previously is used to solve delay differential equations; the delay term is approximated using divided difference interpolation. Numerical results are tabulated and compared with the existing methods. The stability aspects of SDIRK method when applied to DDEs using Lagrange interpolation are investigated and the region of stability is presented.
Runge-Kutta-Nystróm (RKN) method for the solution of special second-order ordinary differential equations of the form ),(yxfy=′′ is discussed. Consequently, Singly Diagonally Implicit Runge-Kutta Nystróm (SDIRKN) method of third-order three stage embedded in fourth-order four stage with small error coefficients is constructed. The stability region of the new method is presented. The method is then used to solve both stiff and non-stiff special second order ODEs and the numerical results suggest that the new method is more efficient compared to the current methods in use.
Finally, introduction to general Runge-Kutta-Nystrom (RKNG) method for the solution of second-order ordinary differential equations of the form ),,(yyxfy′=′′ is given. A new embedded Singly Diagonally Implicit Runge-Kutta-Nystróm General (SDIRKNG) method of third-order four stage embedded in fourth-order five stage is derived. Analysis on the stability aspects of the new method is given and numerical results when the method is used to solve both stiff and non-stiff second order ODEs are presented. The results indicate the superiority of the new method compared to the existing method
Numerical Solution of Ordinary and Delay Differential Equations by Runge-Kutta Type Methods
Runge-Kutta methods for the solution of systems of ordinary differential
equations (ODEs) are described. To overcome the difficulty in implementing fully
implicit Runge-Kutta method and to avoid the limitations of explicit Runge-Kutta
method, we resort to Singly Diagonally Implicit Runge-Kutta (SDIRK) method,
which is computationally efficient and stiffly stable. Consequently, embedded
SDIRK methods of fourth order five stages in fifth order six stages are constructed.
Their regions of stability are presented and numerical results of the methods are
compared with the existing methods.
Stiff systems of ODEs are solved using implicit formulae and require the use
of Newton-like iteration, which needs a lot of computational effort. If the systems can be partitioned dynamically into stiff and nonstiff subsystems then a more
effective code can be developed. Hence, partitioning strategies are discussed in
detail and numerical results based on two techniques to detect stiffness using
SDIRK methods are compared.
A brief introduction to delay differential equations (DDEs) is given. The
stability properties of SDIRK methods, when applied to DDEs, using Lagrange
interpolation to evaluate the delay term, are investigated.
Finally, partitioning strategies for ODEs are adapted to DDEs and numerical
results based on two partitioning techniques, interval wise partitioning and
componentwise partitioning are tabulated and compared
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
Optimization of Nordsieck's Method for the Numerical Integration of Ordinary Differential Equations
Stability and accuracy of Nordsieck's integration method can be improved by choosing the zero-positions of the extraneous roots of the characteristic equation in a suitable way. Optimum zero-positions have been found by minimizing the lower bound of the interval of absolute stability and the coefficient of the truncation error. Various properties of the improved methods have been analysed, such as the behaviour of the high-order terms, the equivalence with multistep methods and the damping of perturbations
Another approach to Runge-Kutta methods
The condition equations are derived by the introduction of a system of equivalent differential equations, avoiding the usual formalism with trees and elementary differentials. Solutions to the condition equations are found by direct optimization, avoiding the necessity to introduce simplifying assumptions upon the Runge-Kutta coefficients. More favourable coefficients, in view of rounding errors, are found
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