46 research outputs found
Construction of Adaptive Multistep Methods for Problems with Discontinuities, Invariants, and Constraints
Adaptive multistep methods have been widely used to solve initial value problems. These ordinary differential equations (ODEs) may arise from semi-discretization of time-dependent partial differential equations(PDEs) or may combine with some algebraic equations to represent a differential algebraic equations (DAEs).In this thesis we study the initialization of multistep methods and parametrize some well-known classesof multistep methods to obtain an adaptive formulation of those methods. The thesis is divided into three main parts; (re-)starting a multistep method, a polynomial formulation of strong stability preserving (SSP)multistep methods and parametric formulation of blocked multistep methods.Depending on the number of steps, a multistep method requires adequate number of initial values tostart the integration. In the view of first part, we look at the available initialization schemes and introduce two family of Runge--Kutta methods derived to start multistep methods with low computational cost and accurate initial values.The proposed starters estimate the error by embedded methods.The second part concerns the variable step-size blocked multistep methods. We use the polynomial formulation of multistep methods applied on ODEs to parametrize blocked multistep methods forthe solution of index-2 Euler-Lagrange DAEs. The performance of the adaptive formulation is verified by some numerical experiments. For the last part, we apply a polynomial formulation of multistep methods to formulate SSP multistep methods that are applied for the solution of semi-discretized hyperbolic PDEs. This formulationallows time adaptivity by construction
Restarting algorithms for simulation problems with discontinuities
Modelica has in its language support for describing discontinuities; so-called events. Modern integrating environments; like Assimulo; provide elaborated event detection and event handling methods. In addition; the overall performance of a simulation of models with discontinuities (hybrid models) depends strongly on methods for restarting integration after event detection. The presented paper reviews two restarting methods; based oRunge--Kutta starters for multistep methods; and presents first experiments on a hybrid system described in Modelica and simulated by JModelica.org/PyFMI and Assimulo
Modified Runge-Kutta methods for solving ODES
A class of Runge-Kutta formulas is examined which permit the calculation of an accurate solution anywhere in the interval of integration. This is used in a code which seldom has to reject a step; rather it takes a reduced step if the estimated error is too large. The absolute stability implications of this are examined
The application of generalized, cyclic, and modified numerical integration algorithms to problems of satellite orbit computation
Generalized, cyclic, and modified multistep numerical integration methods are developed and evaluated for application to problems of satellite orbit computation. Generalized methods are compared with the presently utilized Cowell methods; new cyclic methods are developed for special second-order differential equations; and several modified methods are developed and applied to orbit computation problems. Special computer programs were written to generate coefficients for these methods, and subroutines were written which allow use of these methods with NASA's GEOSTAR computer program
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
Second derivative General Linear Method in Nordsieck form
This paper considers the construction of second derivative general linear methods (SD-GLM) from hybrid LMM and their transformation to Nordsieck
GLM.
How the Runge-Kutta starters for the methods can be derived are given.
The representation of the methods in Nordsieck form has the advantage of easy implementation in variable stepsize.
Starting step size for an ODE solver
AbstractOne of the more critical issues in solving ordinary differential equations by a step-by-step process occurs in the starting phase. Somehow the procedure must be supplied with an initial step size that is on scale for the problem at hand. It must be small enough to yield a reliable solution by the process, but not so small as to significantly affect the efficiency of solution. In this paper, we discuss an algorithm for obtaining a good starting step size and present a subroutine which can be readily used in most ODE solvers
A fifth order runge-kutta RK(5,5) method with error control
In this paper a new Runge-Kutta RK(5, 5) method is introduced. The theory and analysis of its properties are
investigated and compared with the more well known RKF(4, 5) and RK(4, 5) – Merson methods
Analysis of linear and nonlinear stiff problems using the RK-Butcher algorithm
I present a numerical solution of linear and nonlinear stiff
problems using the RK-Butcher algorithm. The obtained discrete
solutions using the RK-Butcher algorithm are found to be very
accurate and are compared with the exact solutions of the linear
and nonlinear stiff problems and also with the Runge-Kutta method
based on arithmetic mean (RKAM). A topic of stability for the
RK-Butcher algorithm is discussed in detail. Error graphs for
discrete and exact solutions are presented in a graphical form to
show the efficiency of the RK-Butcher algorithm. The results
obtained show that RK-Butcher algorithm is more useful for solving
linear and nonlinear stiff problems and the solution can be
obtained for any length of time