3 research outputs found
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
Integrating-factor-based 2-additive Runge-Kutta methods for advection-reaction-diffusion equations
There are three distinct processes that are predominant in models of flowing
media with interacting components: advection, reaction, and diffusion.
Collectively, these processes are typically modelled with partial differential
equations (PDEs) known as advection-reaction-diffusion (ARD) equations.
To solve most PDEs in practice, approximation methods known as numerical methods
are used. The method of lines is used to approximate PDEs with systems of
ordinary differential equations (ODEs) by a process known as
semi-discretization. ODEs are more readily analysed and benefit from
well-developed numerical methods and software. Each term of an ODE that
corresponds to one of the processes of an ARD equation benefits from particular
mathematical properties in a numerical method. These properties are often
mutually exclusive for many basic numerical methods.
A limitation to the widespread use of more complex numerical methods is that the
development of the appropriate software to provide comparisons to existing
numerical methods is not straightforward. Scientific and numerical software is
often inflexible, motivating the development of a class of software known as
problem-solving environments (PSEs). Many existing PSEs such as Matlab have
solvers for ODEs and PDEs but lack specific features, beyond a scripting
language, to readily experiment with novel or existing solution methods. The PSE
developed during the course of this thesis solves ODEs known as initial-value
problems, where only the initial state is fully known. The PSE is used to assess
the performance of new numerical methods for ODEs that integrate each term of a
semi-discretized ARD equation. This PSE is part of the PSE pythODE that uses
object-oriented and software-engineering techniques to allow implementations of
many existing and novel solution methods for ODEs with minimal effort spent on
code modification and integration.
The new numerical methods use a commutator-free exponential Runge-Kutta (CFERK)
method to solve the advection term of an ARD equation. A matrix exponential is
used as the exponential function, but CFERK methods can use other numerical
methods that model the flowing medium. The reaction term is solved separately
using an explicit Runge-Kutta method because solving it along with the
diffusion term can result in stepsize restrictions and hence inefficiency. The
diffusion term is solved using a Runge-Kutta-Chebyshev method that takes
advantage of the spatially symmetric nature of the diffusion process to avoid
stepsize restrictions from a property known as stiffness. The resulting methods,
known as Integrating-factor-based 2-additive Runge-Kutta methods, are shown to be able to find higher-accuracy
solutions in less computational time than competing methods for certain
challenging semi-discretized ARD equations. This demonstrates the practical
viability both of using CFERK methods for advection and a 3-splitting in
general