On the parallel solution of parabolic equations

Parallel algorithms for the solution of linear parabolic problems are proposed. The first of these methods is based on using polynomial approximation to the exponential. It does not require solving any linear systems and is highly parallelizable. The two other methods proposed are based on Pade and Chebyshev approximations to the matrix exponential. The parallelization of these methods is achieved by using partial fraction decomposition techniques to solve the resulting systems and thus offers the potential for increased time parallelism in time dependent problems. Experimental results from the Alliant FX/8 and the Cray Y-MP/832 vector multiprocessors are also presented

Partitioning Techniques and Their Parallelization for Stiff System of Ordinary Differential Equations

A new code based on variable order and variable stepsize component wise partitioning is introduced to solve a system of equations dynamically. In previous partitioning technique researches, once an equation is identified as stiff, it will remain in stiff subsystem until the integration is completed. In this current technique, the system is treated as nonstiff and any equation that caused stiffness will be treated as stiff equation. However, should the characteristics showed the elements of nonstiffness, and then it will be treated again with Adam method. This process will continue switching from stiff to nonstiff vice versa whenever it is necessary until the interval of integration is completed.Next, a block method with R-points generate R new approximate solution values;is a strategy for solving a system and also for parallelizing ODEs. Partitioning this block method to solve stiff differential equations is a new strategy; it is more efficient and takes less computational time compared to the sequential methods. Two partitioning techniques are constructed, Intervalwise Block Partitioning (IBP) and Componentwise Block Partitioning (CBP). Numerical results are compared as validation of its effectiveness. Intervalwise block partitioning will initially treat the systems of equations as nonstiff and solve them using Adams method, by switching to the Backward Differentiation formula when there is a step failure and indication of stiffness. Componentwise block partitioning will place the necessary equations that cause instability and stiffness into the stiff subsystem and solve using Backward Differentiation Formula, while all other equations will still be treated as non-stiff and solved using Adams formula. Parallelizing the partitioning strategies using Message Passing Interface (MPI) is the most appropriate method to solve large system of equations. Parallelizing the right algorithm in the partitioning code will give a better perfonnance with shorter execution times. The graphs of its performance and execution time, visualize the advantages of parallelizing

A 4-Step Implicit Collocation Method for Solution of First and Second Order Odes

ABSTRACT The approach of collocation method approximation will be adopted in the derivation of discrete schemes for direct integration second order ordinary differential equation which are combined together to form a block method. The method is extended to the case in which the approximate solution to a second order (special or general), as well as first order Initial Value Problems(IVPs) can be calculated from the same continuous interpolant and is of order five which is A-stable and has an implicit structure for efficient implementation. The method produces simultaneously approximation of the solution of initial value problems at a block of four points i n x (i=1,2,3,4). Numerical results are given to illustrate the performance method

Transient Stability Simulation by Waveform Relaxation Methods

In this paper, a new methodology for power system dynamic response calculations is presented. The technique known as the waveform relaxation has been extensively used in transient analysis of VLSI circuits and it can take advantage of new architectures in computer systems such as parallel processors. The application in this paper is limited to swing equations of a large power system. Computational results are presented

Block Runge-Kutta Type Method for Direct Integration of Second Order Bvps in Odes

ABSTRACT In this paper, we present the Quade's Type Multistep (QTM) method. The process produces Quade's scheme and some hybrid form which are combined together to form a block method. The method is extended to the case in which the approximate solution to a second order (special or general) Boundary Value Problems(BVPs) can be calculated and is of order six which is A-stable, possesses the Runge-Kutta stability property and has an implicit structure for efficient implementation

An intelligent processing environment for real-time simulation

The development of a highly efficient and thus truly intelligent processing environment for real-time general purpose simulation of continuous systems is described. Such an environment can be created by mapping the simulation process directly onto the University of Alamba's OPERA architecture. To facilitate this effort, the field of continuous simulation is explored, highlighting areas in which efficiency can be improved. Areas in which parallel processing can be applied are also identified, and several general OPERA type hardware configurations that support improved simulation are investigated. Three direct execution parallel processing environments are introduced, each of which greatly improves efficiency by exploiting distinct areas of the simulation process. These suggested environments are candidate architectures around which a highly intelligent real-time simulation configuration can be developed

A bibliography on parallel and vector numerical algorithms

This is a bibliography of numerical methods. It also includes a number of other references on machine architecture, programming language, and other topics of interest to scientific computing. Certain conference proceedings and anthologies which have been published in book form are listed also

Solution of partial differential equations on vector and parallel computers

The present status of numerical methods for partial differential equations on vector and parallel computers was reviewed. The relevant aspects of these computers are discussed and a brief review of their development is included, with particular attention paid to those characteristics that influence algorithm selection. Both direct and iterative methods are given for elliptic equations as well as explicit and implicit methods for initial boundary value problems. The intent is to point out attractive methods as well as areas where this class of computer architecture cannot be fully utilized because of either hardware restrictions or the lack of adequate algorithms. Application areas utilizing these computers are briefly discussed