This thesis presents the Differential Quadrature Method (DQM) for solving time-dependent or heat conduction problem. DQM discretizes the space derivatives giving a system of ordinary differential equations with respect to time and the fourth order Runge Kutta Method (RKM) is employed for solving this system. Stabilities of the ordinary differential equations system and RKM are considered and step sizes are arranged accordingly. The procedure is applied to several time dependent diffusion problems and the solutions are presented in terms of graphics comparing with the exact solutions. This method exhibits high accuracy and efficiency comparing to the other numerical methods.

Mathematical models of electric activity in cardiac tissue are becoming increasingly powerful tools in the study of cardiac arrhythmias. Considered here are mathematical models based on ordinary differential equations (ODEs) and partial differential equations (PDEs) that describe the behaviour of this electrical activity. Generating an efficient numerical solution of these models is a challenging task, and in fact the physiological accuracy of tissue-scale models is often limited by the efficiency of the numerical solution process. In this thesis, we discuss two sets of experiments that test ideas for making the numerical solution process more efficient. In the first set of experiments, we examine the numerical solution of four single cell cardiac electrophysiological models, which consist solely of ODEs. We study the efficiency of using implicit-explicit Runge-Kutta (IMEX-RK) splitting methods to solve these models. We find that variable step-size implementations of IMEX-RK methods (ARK3 and ARK5) that take advantage of Jacobian structure clearly outperform most methods commonly used in practice for two of the models, and they outperform all methods commonly used in practice for the remaining models. In the second set of experiments, we examine the solution of the bidomain model, a model consisting of both ODEs and PDEs that are typically solved separately. We focus these experiments on numerical methods for the solution of the two PDEs in the bidomain model. The most popular method for this task, the Crank-Nicolson method, produces unphysical oscillations; we propose a method based on a second-order L-stable singly diagonally implicit Runge-Kutta (SDIRK) method to eliminate these oscillations.<p> We find that although the SDIRK method is able to eliminate these unphysical oscillations, it is only more efficient for crude error tolerances.

Cylindrical dielectric waveguides such as the optical fiber and photonic crystal fiber are very important passive devices in optical communication systems. There are many kinds of commercial software and methods of simulation at present. In this thesis, we proposed the following four methods to analyze arbitrarily profiled cylindrical dielectric waveguides: The first two methods are modified from published work while the last two methods are entirely developed by ourselves. 1. Cylindrical ABCD matrix method: We take the four continuous electromagnetic field components as main variables and derive the exact four-by-four matrix (with Bessel functions) to relate the four field vector within each homogeneous layer. The electromagnetic field components of the inner and outer layer can propagate toward one of the selected interface of our choice by using the method of ABCD matrix. We can then solve for the £]-value of the waveguide mode with this nonlinear inhomogeneous matrix equation. 2. Runge-Kutta method: Runge-Kutta method is mostly used to solve the initial value problems of the differential equations. In this thesis, we introduce the Runge-Kutta method to solve the first-order four-by-four nonlinear differential equation of the electromagnetic field components and find the £]-value of the cylindrical dielectric waveguides in a similar way depicted in method one. 3. Coupled Ez and Hz method: It uses the axial electromagnetic filed components to solve cylindrical dielectric waveguides. The formulation is similar to cylindrical ABCD matrix method, but it requires less variables then cylindrical ABCD matrix method. The numerical solution obtained from this method is most stable, but it is more complicated to derive harder to write the program. 4. Simple basis expansion method: The simple trigonometric functions (sine or cosine) are chosen as the bases of the horizontal coupled magnetic field equation derived from the second-order differential equation of the transverse magnetic field components. We do not select the horizontal coupling electric field because the normal component of the electric field is discontinuous on the interface. But the normal and tangential components of the magnetic field are continuous across the interfaces. The modal solution problem is converted to a linear matrix eigenvalue-eigenvector equation which is solved by the standard linear algebra routines. We will compare these four numerical methods with one another. The characteristics and advantage as well as the disadvantage of each method will be studied and compared in detail.

In munitions technologies, hazard investigations for explosive (or more generally energetic material) including systems is a very important issue to achieve insensitivity. Determining the response of energetic materials to different types of mechanical or thermal threats has vital importance to achieve an effective and safe munitions design and since 1970&rsquo / s, lots of studies have been performed in this research field to simulate the dynamic response of energetic materials under some circumstances. The testing for hazard investigations is a very expensive and dangerous topic in munitions design studies. Therefore, especially in conceptual design phase, the numerical simulation tools for hazard investigations has been used by ballistic researchers since 1970s. The main modeling approach in such simulation tools is the numerical simulation of deflagration-todetonation transition (DDT) phenomenon. By this motivation, in this thesis study, the numerical simulation of DDT phenomenon in solid energetic materials which occurs under some mechanical effects is performed. One dimensional and two dimensional solvers are developed by using some well-known models defined in open literature for HMX (C4 H8 N8 O8) with 73 % particle load which is a typical granular, energetic, solid, explosive ingredient. These models include the two-phase conservation equations coupled with the combustion, interphase drag interaction, interphase heat transfer interaction and compaction source terms. In the developed solvers, the governing partial differential equation (PDE) system is solved by employing high-order central differences for time and spatial integration. The two-dimensional solver is developed by extending the complete two-phase model of the one-dimensional solver without any reductions in momentum and energy conservation equations. In one dimensional calculations, compaction, ignition, deflagration and transition to detonation characteristics are investigated and, a good agreement is achieved with the open literature. In two dimensional calculations, effect of blunt and sharp-nosed projectile impact situations on compaction and ignition characteristics of a typical explosive bed is investigated. A minimum impact velocity under which ignition in the domain fails is sought. Then the developed solver is tested with a special wave-shaper problem and the results are in a good agreement with those of a commercial software.

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.

No description available.

An initial-value problem consists of an ordinary differential equation subject to an initial condition. The right-hand side of the differential equation can be interpreted as additively split when it is comprised of the sum of two or more contributing factors. For instance, the right-hand sides of initial-value problems derived from advection-diffusion-reaction equations are comprised of the sum of terms emanating from three distinct physical processes: advection, diffusion, and reaction. In some cases, solutions to initial-value problems can be calculated analytically, but when an analytic solution is unknown or nonexistent, methods of numerical integration are used to calculate solutions. The runtime performance of numerical methods is problem dependent; therefore, one must choose an appropriate numerical method to achieve favourable performance, according to characteristics of the problem. Additive methods of numerical integration apply distinct methods to the distinct contributing factors of an additively split problem. Treating the contributing factors with methods that are known to perform well on them individually has the potential to yield an additive method that outperforms single methods applied to the entire (unsplit) problem. Splittings of the right-hand side can be physics-based, i.e., based on physical characteristics of the problem, such as advection, diffusion, or reaction terms. Splittings can also be based on linearization, called Jacobian splitting in this thesis, where the linearized part of the problem is treated with one method and the rest of the problem is treated with another. A comparison of these splitting techniques is performed by applying a set of additive methods to a test suite of problems. Many common non-additive methods are also included to serve as a performance baseline. To perform this numerical study, a problem-solving environment was developed to evaluate permutations of problems, methods, and their associated parameters. The test suite is comprised of several distinct advection-diffusion-reaction equations that have been chosen to represent a wide range of common problem characteristics. When solving split problems in the test suite, it is found that additive Runge–Kutta methods of orders three, four, and five using Jacobian splitting generally outperform those same methods using physics-based splitting. These results provide evidence that Jacobian splitting is an effective approach when solving such initial-value problems in practice.

No description available.

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.

This work is devoted to the convergence analysis of a modified Runge-Kutta-type iterative regularization method for solving nonlinear ill-posed problems under a priori and a posteriori stopping rules. The convergence rate results of the proposed method can be obtained under Hölder-type source-wise condition if the Fréchet derivative is properly scaled and locally Lipschitz continuous. Numerical results are achieved by using the Levenberg-Marquardt and Radau methods.