Thesis (Ph. D.)--Oregon State University, 1974. / Typescript (photocopy). Includes bibliographical references. Also available on the World Wide Web.

This paper investigates the Runge-Kutta method of numerically integrating ordinary differential equations. An existence theorem is given insuring a solution to the differential equation, then the theorem is modified to yield an analytic solution. The derivation of the method itself is followed by an analysis of the inherent error.

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Simulation of large physical systems often leads to initial value problems in which some of the solution components contain high frequency oscillations and/or fast transients, while the remaining solution components are relatively slowly varying. Such a system is referred to as two-time-scale (TTS), which is a partial generalization of the concept of stiffness. When using conventional numerical techniques for integration of TTS systems, the rapidly varying components dictate the use of small stepsizes, with the result that the slowly varying components are integrated very inefficiently. This could mean that the computer time required for integration is excessive. To overcome this difficulty, the system is partitioned into "fast" and "slow" subsystems, containing the rapidly and slowly varying components of the solution respectively. Integration is then performed using small stepsizes for the fast subsystem and relatively large stepsizes for the slow subsystem. This is referred to as multirate integration, and it can lead to substantial savings in computer time required for integration of large systems having relatively few fast solution components. This study is devoted to multirate integration of TTS initial value problems which are partitioned into fast and slow subsystems. Techniques for partitioning are not considered here. Multirate integration algorithms based on explicit Runge-Kutta (RK) methods are developed. Such algorithms require a means for communication between the subsystems. Internally embedded RK methods are introduced to aid in computing interpolated values of the slow variables, which are supplied to the fast subsystem. The use of averaging in the fast subsystem is discussed in connection with communication from the fast to the slow subsystem. Theoretical support for this is presented in a special case. A proof of convergence is given for a multirate algorithm based on Euler's method. Absolute stability of this algorithm is also discussed. Four multirate integration routines are presented. Two of these are based on a fixed-step fourth order RK method, and one is based on the variable step Runge-Kutta-Merson scheme. The performance of these routines is compared to that of several other integration schemes, including Gear's method and Hindmarsh's EPISODE package. For this purpose, both linear and nonlinear examples are presented. It is found that multirate techniques show promise for linear systems having eigenvalues near the imaginary axis. Such systems are known to present difficulty for Gear's method and EPISODE. A nonlinear TTS model of an autopilot is presented. The variable step multirate routine is found to be substantially more efficient for this example than any other method tested. Preliminary results are also included for a pressurized water reactor model. Indications are that multirate techniques may prove fruitful for this model. Lastly, an investigation of the effects of the step-size ratio (between subsystems) is included. In addition, several suggestions for further work are given, including the possibility of using multistep methods for integration of the slow subsystem.

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In this work, we present an amplitude-shape method for solving evolution problems described by partial differential equations. The method is capable of recognizing the special structure of many evolution problems. In particular, the stiff system of ordinary differential equations resulting from the semi-discretization of partial differential equations is considered. The method involves transforming the system so that only a few equations are stiff and the majority of the equations remain non-stiff. The system is treated with a mixed explicit-implicit scheme with a built-in error control mechanism. This approach proved to be very effective for the solution of stiff systems of equations describing spatially dependent chemical kinetics. / Thesis (Ph.D.)-University of Natal, 1997.

Made available in DSpace on 2016-02-05T18:29:18Z (GMT). No. of bitstreams: 0 Previous issue date: 2015-10-09. Added 1 bitstream(s) on 2016-02-05T18:33:03Z : No. of bitstreams: 1 000857260.pdf: 965706 bytes, checksum: 4793a2f9effb16b97b2baad9ae919dd7 (MD5) / O propósito deste trabalho é explorar os métodos numéricos de passo único, denominados métodos de Euler e Runge - Kutta, e os métodos de passos múltiplos, denominados métodos de Adams-Bashforth e Adams-Moulton, para encontrar soluções aproximadas de problemas de valor inicial para equações diferenciais ordinárias de primeira ordem / The goal of this work is to explore the one-step numerical methods, called methods of Euler and Runge - Kutta, and the multistep numerical methods, called methods of Adams-Bashforth and Adams Moulton, for finding approximate solutions of initial value problems for first-order ordinary differential equations

Earthquake ground motion can induce out-of-phase vibrations between adjacent structures due to differences in dynamic characteristics, which can result in impact or pounding of the structures if the at-rest separation is insufficient to accommodate the relative displacements. In bridges, seismic pounding between adjacent decks or between deck and abutment can result in localized deck damage, bearing failure, damage to shear keys and abutments, and even contribute to the collapse of bridge spans. This study investigates pounding in bridges from an analytical perspective. A simplified nonlinear model of a multiple-frame bridge is developed in MATLAB incorporating the effects of inelastic frame action, nonlinear hinge behavior and abutments. The equations of motion of the bridge response to longitudinal ground excitation are assembled and solved using the fourth-order Runge-Kutta method. Pounding is simulated using contact force-based models such as the linear spring, Kelvin and Hertz models, as well as the momentum-based stereomechanical method. In addition, a Hertz contact model with nonlinear damping (Hertzdamp model) is also introduced to model impact. The primary factors controlling the pounding response are identified as the frame period ratio, ground motion effective period ratio, restrainer stiffness ratio and frame ductility ratio. Pounding is most critical for highly out-of-phase frames. Impact models without energy dissipation overestimate the stiff system displacements by 15%-25% for highly out-of-phase, elastic systems experiencing moderate to strong ground excitation. The Hertzdamp model is found to be the most effective in representing impact. Traditional column hysteresis models such as the elasto-plastic and bilinear models underestimate the stiff system amplification and overestimate the flexible system amplification due to impact, when compared with stiffness and strength degrading models. Strength degradation and pounding are critical on the stiff system response to near field ground motions, for highly out-of-phase systems. Current design procedures are adequate in capturing the nonlinear hinge response when the bridge columns are elastic, but require revisions such as the introduction of time dependent reduction factors, and a frame design period to work for inelastic situations. Finally, a bilinear truss element with a gap is proposed for implementing energy dissipating impact models in commercial structural software.