49,291 research outputs found
An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method
In this paper we propose a collocation method for solving some well-known
classes of Lane-Emden type equations which are nonlinear ordinary differential
equations on the semi-infinite domain. They are categorized as singular initial
value problems. The proposed approach is based on a Hermite function
collocation (HFC) method. To illustrate the reliability of the method, some
special cases of the equations are solved as test examples. The new method
reduces the solution of a problem to the solution of a system of algebraic
equations. Hermite functions have prefect properties that make them useful to
achieve this goal. We compare the present work with some well-known results and
show that the new method is efficient and applicable.Comment: 34 pages, 13 figures, Published in "Computer Physics Communications
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
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
ИНТЕЛЛЕКТУАЛЬНЫЙ числовым программным ДЛЯ MIMD-компьютер
For most scientific and engineering problems simulated on computers the solving of problems of the computational mathematics with approximately given initial data constitutes an intermediate or a final stage. Basic problems of the computational mathematics include the investigating and solving of linear algebraic systems, evaluating of eigenvalues and eigenvectors of matrices, the solving of systems of non-linear equations, numerical integration of initial- value problems for systems of ordinary differential equations.Для більшості наукових та інженерних задач моделювання на ЕОМ рішення задач обчислювальної математики з наближено заданими вихідними даними складає проміжний або остаточний етап. Основні проблеми обчислювальної математики відносяться дослідження і рішення лінійних алгебраїчних систем оцінки власних значень і власних векторів матриць, рішення систем нелінійних рівнянь, чисельного інтегрування початково задач для систем звичайних диференціальних рівнянь.Для большинства научных и инженерных задач моделирования на ЭВМ решение задач вычислительной математики с приближенно заданным исходным данным составляет промежуточный или окончательный этап. Основные проблемы вычислительной математики относятся исследования и решения линейных алгебраических систем оценки собственных значений и собственных векторов матриц, решение систем нелинейных уравнений, численного интегрирования начально задач для систем обыкновенных дифференциальных уравнений
Model Problems in Numerical Stability Theory for Initial Value Problems
In the past numerical stability theory for initial value problems in ordinary differential equations has been dominated by the study of problems with simple dynamics; this has been motivated by the need to study error propagation mechanisms in stiff problems, a question modeled effectively by contractive linear or nonlinear problems. While this has resulted in a coherent and self-contained body of knowledge, it has never been entirely clear to what extent this theory is relevant for problems exhibiting more complicated dynamics. Recently there have been a number of studies of numerical stability for wider classes of problems admitting more complicated dynamics. This on-going work is unified and, in particular, striking similarities between this new developing stability theory and the classical linear and nonlinear stability theories are emphasized.
The classical theories of A, B and algebraic stability for Runge–Kutta methods are briefly reviewed; the dynamics of solutions within the classes of equations to which these theories apply—linear decay and contractive problems—are studied. Four other categories of equations—gradient, dissipative, conservative and Hamiltonian systems—are considered. Relationships and differences between the possible dynamics in each category, which range from multiple competing equilibria to chaotic solutions, are highlighted. Runge-Kutta schemes that preserve the dynamical structure of the underlying problem are sought, and indications of a strong relationship between the developing stability theory for these new categories and the classical existing stability theory for the older problems are given. Algebraic stability, in particular, is seen to play a central role.
It should be emphasized that in all cases the class of methods for which a coherent and complete numerical stability theory exists, given a structural assumption on the initial value problem, is often considerably smaller than the class of methods found to be effective in practice. Nonetheless it is arguable that it is valuable to develop such stability theories to provide a firm theoretical framework in which to interpret existing methods and to formulate goals in the construction of new methods. Furthermore, there are indications that the theory of algebraic stability may sometimes be useful in the analysis of error control codes which are not stable in a fixed step implementation; this work is described
Optimization of chemical plant simulation using double collocation
A method has been constructed for the solution of a wide range of chemical plant simulation models including differential equations and optimization. Double orthogonal collocation on finite elements is applied to convert the model into an NLP problem that is solved either by the VF 13AD package based on successive quadratic programming, or by the GRG2 package, based on the generalized reduced gradient method. This approach is termed simultaneous optimization and solution strategy. The objective functional can contain integral terms. The state and control variables can have time delays. Equalities and inequalities containing state and control variables can be included into the model as well as algebraic equations and inequalities. The maximum number of independent variables is 2. Problems containing 3 independent variables can be transformed into problems having 2 independent variables using finite differencing. The maximum number of NLP variables and constraints is 1500. The method is also suitable for solving ordinary and partial differential equations. The state functions are approximated by a linear combination of Lagrange interpolation polynomials. The control function can either be approximated by a linear combination of Lagrange interpolation polynomials or by a piecewise constant function over finite elements. The number of internal collocation points can vary by finite elements. The residual error is evaluated at arbitrarily chosen equidistant grid-points, thus enabling the user to check the accuracy of the solution between collocation points, where the solution is exact. The solution functions can be tabulated. There is an option to use control vector parameterization to solve optimization problems containing initial value ordinary differential equations. When there are many differential equations or the upper integration limit should be selected optimally then this approach should be used. The portability of the package has been addressed converting the package from V AX FORTRAN 77 into IBM PC FORTRAN 77 and into SUN SPARC 2000 FORTRAN 77. Computer runs have shown that the method can reproduce optimization problems published in the literature. The GRG2 and the VF I 3AD packages, integrated into the optimization package, proved to be robust and reliable. The package contains an executive module, a module performing control vector parameterization and 2 nonlinear problem solver modules, GRG2 and VF I 3AD. There is a stand-alone module that converts the differential-algebraic optimization problem into a nonlinear programming problem
Converting DAE models to ODE models: application to reactive Rayleigh distillation
This paper illustrates the application of an index reduction method to some differential algebraic equations
(DAE) modelling the reactive Rayleigh distillation. After two deflation steps, this DAE is converted to an
equivalent first-order explicit ordinary differential equation (ODE). This ODE involves a reduced number of
dependent variables, and some evaluations of implicit functions defined, either from the original algebraic
constraints, or from the hidden ones. Consistent initial conditions are no longer to be computed; at the
opposite of some other index reduction methods, which generate a drift-off effect, the algebraic constraints
remain satisfied at any time; and, finally, the computational effort to solve the ODE may be less than the
one associated to the original DAE
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