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

The effect of FPU architecture on a dynamic precision algorithm for the solution of differential equations

Solution of lnitial Value Problems (IVPs) is an important application in scientific computing. Methods for solving these problems use techniques for reducing the error and increasing the speed of the computation. This paper introduces a class of algorithms which dynamically reconfigure their operating parameters to reduce the computation time. By dynamically varying the precision of the arithmetic being performed, it is possible to obtain dramatic speedups on certain architectures when solving IVPs. This paper illustrates how various architectures impact on a dynamic precision version of the Runge-Kutta-Fehlberg algorithm. It is shown that a speedup of over 30 percent is possible for both massively parallel processors and vector supercomputers

Systematic construction of efficient six-stage fifth-order explicit Runge-Kutta embedded pairs without standard simplifying assumptions

This thesis examines methodologies and software to construct explicit Runge-Kutta (ERK) pairs for solving initial value problems (IVPs) by constructing efficient six-stage fifth-order ERK pairs without standard simplifying assumptions. The problem of whether efficient higher-order ERK pairs can be constructed algebraically without the standard simplifying assumptions dates back to at least the 1960s, with Cassity's complete solution of the six-stage fifth-order order conditions. Although RK methods based on the six-stage fifth-order order conditions have been widely studied and have continuing practical importance, prior to this thesis, the aforementioned complete solution to these order conditions has no published usage beyond the original series of publications by Cassity in the 1960s. The complete solution of six-stage fifth-order ERK order conditions published by Cassity in 1969 is not in a formulation that can easily be used for practical purposes, such as a software implementation. However, it is shown in this thesis that when the order conditions are solved and formulated appropriately using a computer algebra system (CAS), the generated code can be used for practical purposes and the complete solution is readily extended to ERK pairs. The condensed matrix form of the order conditions introduced by Cassity in 1969 is shown to be an ideal methodology, which probably has wider applicability, for solving order conditions using a CAS. The software package OCSage developed for this thesis, in order to solve the order conditions and study the properties of the resulting methods, is built on top of the Sage CAS. However, in order to effectively determine that the constructed ERK pairs without standard simplifying assumptions are in fact efficient by some well-defined criteria, the process of selecting the coefficients of ERK pairs is re-examined in conjunction with a sufficient amount of performance data. The pythODE software package developed for this thesis is used to generate a large amount of performance data from a large selection of candidate ERK pairs found using OCSage. In particular, it is shown that there is unlikely to be a well-defined methodology for selecting optimal pairs for general-purpose use, other than avoiding poor choices of certain properties and ensuring the error coefficients are as small as possible. However, for IVPs from celestial mechanics, there are obvious optimal pairs that have specific values of a small subset of the principal error coefficients (PECs). Statements seen in the literature that the best that can be done is treating all PECs equally do not necessarily apply to at least some broad classes of IVPs. By choosing ERK pairs based on specific values of individual PECs, not only are ERK pairs that are 20-30% more efficient than comparable published pairs found for test sets of IVPs from celestial mechanics, but the variation in performance between the best and worst ERK pairs that otherwise would seem to have similar properties is reduced from a factor of 2 down to as low as 15%. Based on observations of the small number of IVPs of other classes in common IVP test sets, there are other classes of IVPs that have different optimal values of the PECs. A more general contribution of this thesis is that it specifically demonstrates how specialized software tools and a larger amount of performance data than is typical can support novel empirical insights into numerical methods

Using SIMD and SIMT vectorization to evaluate sparse chemical kinetic Jacobian matrices and thermochemical source terms

Accurately predicting key combustion phenomena in reactive-flow simulations, e.g., lean blow-out, extinction/ignition limits and pollutant formation, necessitates the use of detailed chemical kinetics. The large size and high levels of numerical stiffness typically present in chemical kinetic models relevant to transportation/power-generation applications make the efficient evaluation/factorization of the chemical kinetic Jacobian and thermochemical source-terms critical to the performance of reactive-flow codes. Here we investigate the performance of vectorized evaluation of constant-pressure/volume thermochemical source-term and sparse/dense chemical kinetic Jacobians using single-instruction, multiple-data (SIMD) and single-instruction, multiple thread (SIMT) paradigms. These are implemented in pyJac, an open-source, reproducible code generation platform. Selected chemical kinetic models covering the range of sizes typically used in reactive-flow simulations were used for demonstration. A new formulation of the chemical kinetic governing equations was derived and verified, resulting in Jacobian sparsities of 28.6-92.0% for the tested models. Speedups of 3.40-4.08 x were found for shallow-vectorized OpenCL source-rate evaluation compared with a parallel OpenMP code on an avx2 central processing unit (CPU), increasing to 6.63-9.44 x and 3.03-4.23 x for sparse and dense chemical kinetic Jacobian evaluation, respectively. Furthermore, the effect of data-ordering was investigated and a storage pattern specifically formulated for vectorized evaluation was proposed; as well, the effect of the constant pressure/volume assumptions and varying vector widths were studied on source-term evaluation performance. Speedups reached up to 17.60 x and 45.13 x for dense and sparse evaluation on the GPU, and up to 55.11 x and 245.63 x on the CPU over a first-order finite-difference Jacobian approach. Further, dense Jacobian evaluation was up to 19.56 x and 2.84 x times faster than a previous version of pyJac on a CPU and GPU, respectively. Finally, future directions for vectorized chemical kinetic evaluation and sparse linear-algebra techniques were discussed. (C) 2018 The Combustion Institute. Published by Elsevier Inc. All rights reserved