This thesis deals with the direct simulation Monte Carlo (DSMC) method of analysing gas flows. The DSMC method was initially proposed as a method for predicting rarefied flows where the Navier-Stokes equations are inaccurate. It has now been extended to near continuum flows. The method models gas flows using simulation molecules which represent a large number of real molecules in a probabilistic simulation to solve the Boltzmann equation. Molecules are moved through a simulation of physical space in a realistic manner that is directly coupled to physical time such that unsteady flow characteristics are modelled. Intermolecular collisions and moleculesurface collisions are calculated using probabilistic, phenomenological models. The fundamental assumption of the DSMC method is that the molecular movement and collision phases can be decoupled over time periods that are smaller than the mean collision time. Two obstacles to the wide spread use of the DSMC method as an engineering tool are in the areas of simulation configuration, which is the configuration of the simulation parameters to provide a valid solution, and the time required to obtain a solution. For complex problems, the simulation will need to be run multiple times, with the simulation configuration being modified between runs to provide an accurate solution for the previous run�s results, until the solution converges. This task is time consuming and requires the user to have a good understanding of the DSMC method. Furthermore, the computational resources required by a DSMC simulation increase rapidly as the simulation approaches the continuum regime. Similarly, the computational requirements of three-dimensional problems are generally two orders of magnitude more than two-dimensional problems. These large computational requirements significantly limit the range of problems that can be practically solved on an engineering workstation or desktop computer. The first major contribution of this thesis is in the development of a DSMC implementation that automatically adapts the simulation. Rather than modifying the simulation configuration between solution runs, this thesis presents the formulation of algorithms that allow the simulation configuration to be automatically adapted during a single run. These adaption algorithms adjust the three main parameters that effect the accuracy of a DSMC simulation, namely the solution grid, the time step and the simulation molecule number density. The second major contribution extends the parallelisation of the DSMC method. The implementation developed in this thesis combines the capability to use a cluster of computers to increase the maximum size of problem that can be solved while simultaneously allowing excess computational resources to decrease the total solution time. Results are presented to verify the accuracy of the underlying DSMC implementation, the utility of the solution adaption algorithms and the efficiency of the parallelisation implementation.

This thesis deals with the direct simulation Monte Carlo (DSMC) method of analysing gas flows. The DSMC method was initially proposed as a method for predicting rarefied flows where the Navier-Stokes equations are inaccurate. It has now been extended to near continuum flows. The method models gas flows using simulation molecules which represent a large number of real molecules in a probabilistic simulation to solve the Boltzmann equation. Molecules are moved through a simulation of physical space in a realistic manner that is directly coupled to physical time such that unsteady flow characteristics are modelled. Intermolecular collisions and moleculesurface collisions are calculated using probabilistic, phenomenological models. The fundamental assumption of the DSMC method is that the molecular movement and collision phases can be decoupled over time periods that are smaller than the mean collision time. Two obstacles to the wide spread use of the DSMC method as an engineering tool are in the areas of simulation configuration, which is the configuration of the simulation parameters to provide a valid solution, and the time required to obtain a solution. For complex problems, the simulation will need to be run multiple times, with the simulation configuration being modified between runs to provide an accurate solution for the previous run�s results, until the solution converges. This task is time consuming and requires the user to have a good understanding of the DSMC method. Furthermore, the computational resources required by a DSMC simulation increase rapidly as the simulation approaches the continuum regime. Similarly, the computational requirements of three-dimensional problems are generally two orders of magnitude more than two-dimensional problems. These large computational requirements significantly limit the range of problems that can be practically solved on an engineering workstation or desktop computer. The first major contribution of this thesis is in the development of a DSMC implementation that automatically adapts the simulation. Rather than modifying the simulation configuration between solution runs, this thesis presents the formulation of algorithms that allow the simulation configuration to be automatically adapted during a single run. These adaption algorithms adjust the three main parameters that effect the accuracy of a DSMC simulation, namely the solution grid, the time step and the simulation molecule number density. The second major contribution extends the parallelisation of the DSMC method. The implementation developed in this thesis combines the capability to use a cluster of computers to increase the maximum size of problem that can be solved while simultaneously allowing excess computational resources to decrease the total solution time. Results are presented to verify the accuracy of the underlying DSMC implementation, the utility of the solution adaption algorithms and the efficiency of the parallelisation implementation.

Abstract Several numerical methods and algorithms, for solving the mathematical model of a continuous casting process, are presented, and theoretically studied, in this work. The numerical algorithms can be divided in to three different groups: the Schwarz type overlapping methods, the nonoverlapping Splitting iterative methods, and the Predictor-Corrector type nonoverlapping methods. These algorithms are all so-called parallel algorithms i.e., they are highly suitable for parallel computers. Multiplicative, additive Schwarz alternating method and two asynchronous domain decomposition methods, which appear to be a two-stage Schwarz alternating algorithms, are theoretically and numerically studied. Unique solvability of the fully implicit and semi-implicit finite difference schemes as well as monotone dependence of the solution on the right-hand side are proved. Geometric rate of convergence for the iterative methods is investigated. Splitting iterative methods for the sum of maximal monotone and single-valued monotone operators in a finite-dimensional space are studied. Convergence, rate of convergence and optimal iterative parameters are derived. A two-stage iterative method with inner iterations is analyzed in the case when both operators are linear, self-adjoint and positive definite. Several new finite-difference schemes for a nonlinear convection-diffusion problem are constructed and numerically studied. These schemes are constructed on the basis of non-overlapping domain decomposition and predictor-corrector approach. Different non-overlapping decompositions of a domain, with cross-points and angles, schemes with grid refinement in time in some subdomains, are used. All proposed algorithms are extensively numerically tested and are founded stable and accurate under natural assumptions for time and space grid steps. The advantages and disadvantages of the numerical methods are clearly seen in the numerical examples. All of the algorithms presented are quite easy and straight forward, from an implementation point of view. The speedups show that splitting iterative method can be parallelized better than multiplicative or additive Schwarz alternating method. The numerical examples show that the multidecomposition method is a very effective numerical method for solving the continuous casting problem. The idea of dividing the subdomains to smaller subdomains seems to be very beneficial and profitable. The advantages of multidecomposition methods over other methods is obvious. Multidecomposition methods are extremely quick, while being just as accurate as other methods. The numerical results for one processor seem to be very promising.

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.