244 research outputs found
STICAP: A linear circuit analysis program with stiff systems capability. Volume 1: Theory manual
STICAP (Stiff Circuit Analysis Program) is a FORTRAN 4 computer program written for the CDC-6400-6600 computer series and SCOPE 3.0 operating system. It provides the circuit analyst a tool for automatically computing the transient responses and frequency responses of large linear time invariant networks, both stiff and nonstiff (algorithms and numerical integration techniques are described). The circuit description and user's program input language is engineer-oriented, making simple the task of using the program. Engineering theories underlying STICAP are examined. A user's manual is included which explains user interaction with the program and gives results of typical circuit design applications. Also, the program structure from a systems programmer's viewpoint is depicted and flow charts and other software documentation are given
Degenerate Variational Integrators for Magnetic Field Line Flow and Guiding Center Trajectories
Symplectic integrators offer many advantages for the numerical solution of
Hamiltonian differential equations, including bounded energy error and the
preservation of invariant sets. Two of the central Hamiltonian systems
encountered in plasma physics --- the flow of magnetic field lines and the
guiding center motion of magnetized charged particles --- resist symplectic
integration by conventional means because the dynamics are most naturally
formulated in non-canonical coordinates, i.e., coordinates lacking the familiar
partitioning. Recent efforts made progress toward non-canonical
symplectic integration of these systems by appealing to the variational
integration framework; however, those integrators were multistep methods and
later found to be numerically unstable due to parasitic mode instabilities.
This work eliminates the multistep character and, therefore, the parasitic mode
instabilities via an adaptation of the variational integration formalism that
we deem ``degenerate variational integration''. Both the magnetic field line
and guiding center Lagrangians are degenerate in the sense that their resultant
Euler-Lagrange equations are systems of first-order ODEs. We show that
retaining the same degree of degeneracy when constructing a discrete Lagrangian
yields one-step variational integrators preserving a non-canonical symplectic
structure on the original Hamiltonian phase space. The advantages of the new
algorithms are demonstrated via numerical examples, demonstrating superior
stability compared to existing variational integrators for these systems and
superior qualitative behavior compared to non-conservative algorithms
Some investigations into the numerical solution of initial value problems in ordinary differential equations
PhD ThesisIn this thesis several topics in the numerical solution
of the initial value problem in first-order ordinary diff'erentlal
equations are investigated,
Consideration is given initially to stiff differential
equations and their solution by stiffly-stable linear multistep
methods which incorporate second derivative terms. Attempts are
made to increase the size of the stability regions for these
methods both by particular choices for the third characteristic
polynomial and by the use of optimization techniques while
investigations are carried out regarding the capabilities of a
high order method.
Subsequent work is concerned with the development of
Runge-Kutta methods which include second-derivative terms and
are implicit with respect to y rather than k. Methods of
order three and four are proposed which are L-stable.
The major part of the thesis is devoted to the establishment
of recurrence relations for operators associated with linear
multistep methods which are based on a non-polynomial
representation of the theoretical solution. A complete set of
recurrence relations is developed for both implicit and
explicit multistep methods which are based on a representation
involving a polynomial part and any number of arbitrary functions.
The amount of work involved in obtaining mulc iste, :ne::l'lJds by this
technique is considered and criteria are proposed to Jecide when
this particular method of derivation should be em~loyed.
The thesis is conclud~d by using Prony's method to develop
one-step methods and multistep methods which are exponentially
adaptive and as such can be useful in obtaining solutions to
problems which are exponential in nature
Collocation methods for a class of second order initial value problems with oscillatory solutions
We derive and analyse two families of multistep collocation methods for periodic initial-value problems of the form y" = f(x, y); y((^x)o) = yo, y(^1)(xo) = zo involving ordinary differential equations of second order in which the first derivative does not appear explicitly. A survey of recent results and proposed numerical methods is given in chapter 2. Chapter 3 is devoted to the analysis of a family of implicit Chebyshev methods proposed by Panovsky k Richardson. We show that for each non-negative integer r, there are two methods of order 2r from this family which possess non-vanishing intervals of periodicity. The equivalence of these methods with one-step collocation methods is also established, and these methods are shown to be neither P-stable nor symplectic. In chapters 4 and 5, two families of multistep collocation methods are derived, and their order and stability properties are investigated. A detailed analysis of the two-step symmetric methods from each class is also given. The multistep Runge-Kutta-Nystrom methods of chapter 4 are found to be difficult to analyse, and the specific examples considered are found to perform poorly in the areas of both accuracy and stability. By contrast, the two-step symmetric hybrid methods of chapter 5 are shown to have excellent stability properties, in particular we show that all two-step 27V-point methods of this type possess non-vanishing intervals of periodicity, and we give conditions under which these methods are almost P-stable. P-stable and efficient methods from this family are obtained and demonstrated in numerical experiments. A simple, cheap and effective error estimator for these methods is also given
High-order finite difference schemes for the solution of second-order BVPs
AbstractWe introduce new methods in the class of boundary value methods (BVMs) to solve boundary value problems (BVPs) for a second-order ODE. These formulae correspond to the high-order generalizations of classical finite difference schemes for the first and second derivatives. In this research, we carry out the analysis of the conditioning and of the time-reversal symmetry of the discrete solution for a linear convection–diffusion ODE problem. We present numerical examples emphasizing the good convergence behavior of the new schemes. Finally, we show how these methods can be applied in several space dimensions on a uniform mesh
Increasing the real stability boundary of explicit methods
AbstractBased on the simplest well-known integration rules (such as the forward Euler scheme and the “classical” Runge-Kutta method), an extension is proposed to enlarge the real stability boundary. The main characteristic of the resulting schemes is that the computational complexity is hardly increased
Recommended from our members
SciCADE 95: International conference on scientific computation and differential equations
This report consists of abstracts from the conference. Topics include algorithms, computer codes, and numerical solutions for differential equations. Linear and nonlinear as well as boundary-value and initial-value problems are covered. Various applications of these problems are also included
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