875 research outputs found
Construction of additive semi-implicit Runge-Kutta methods with low-storage requirements
The final publication is available at Springer via
http://dx.doi.org/ 10.1007/s10915-015-0116-2Space discretization of some time-dependent partial differential equations gives rise to systems of
ordinary differential equations in additive form whose terms have different stiffness properties. In these
cases, implicit methods should be used to integrate the stiff terms while efficient explicit methods can be
used for the non-stiff part of the problem. However, for systems with a large number of equations, memory
storage requirement is also an important issue. When the high dimension of the problem compromises
the computer memory capacity, it is important to incorporate low memory usage to some other properties of the scheme.
In this paper we consider Additive Semi-Implicit Runge-Kutta (ASIRK) methods, a class of implicitexplicit
Runge-Kutta methods for additive differential systems. We construct two second order 3-stage
ASIRK schemes with low-storage requirements. Having in mind problems with stiffness parameters,
besides accuracy and stability properties, we also impose stiff accuracy conditions. The numerical experiments done show the advantages of the new methods.Supported by Ministerio de Economía y Competividad, project MTM2011-23203
Implicit and Implicit-Explicit Strong Stability Preserving Runge-Kutta Methods with High Linear Order
When evolving in time the solution of a hyperbolic partial differential
equation, it is often desirable to use high order strong stability preserving
(SSP) time discretizations. These time discretizations preserve the
monotonicity properties satisfied by the spatial discretization when coupled
with the first order forward Euler, under a certain time-step restriction.
While the allowable time-step depends on both the spatial and temporal
discretizations, the contribution of the temporal discretization can be
isolated by taking the ratio of the allowable time-step of the high order
method to the forward Euler time-step. This ratio is called the strong
stability coefficient. The search for high order strong stability time-stepping
methods with high order and large allowable time-step had been an active area
of research. It is known that implicit SSP Runge-Kutta methods exist only up to
sixth order. However, if we restrict ourselves to solving only linear
autonomous problems, the order conditions simplify and we can find implicit SSP
Runge-Kutta methods of any linear order. In the current work we aim to find
very high linear order implicit SSP Runge-Kutta methods that are optimal in
terms of allowable time-step. Next, we formulate an optimization problem for
implicit-explicit (IMEX) SSP Runge-Kutta methods and find implicit methods with
large linear stability regions that pair with known explicit SSP Runge-Kutta
methods of orders plin=3,4,6 as well as optimized IMEX SSP Runge-Kutta pairs
that have high linear order and nonlinear orders p=2,3,4. These methods are
then tested on sample problems to verify order of convergence and to
demonstrate the sharpness of the SSP coefficient and the typical behavior of
these methods on test problems
Review of Summation-by-parts schemes for initial-boundary-value problems
High-order finite difference methods are efficient, easy to program, scales
well in multiple dimensions and can be modified locally for various reasons
(such as shock treatment for example). The main drawback have been the
complicated and sometimes even mysterious stability treatment at boundaries and
interfaces required for a stable scheme. The research on summation-by-parts
operators and weak boundary conditions during the last 20 years have removed
this drawback and now reached a mature state. It is now possible to construct
stable and high order accurate multi-block finite difference schemes in a
systematic building-block-like manner. In this paper we will review this
development, point out the main contributions and speculate about the next
lines of research in this area
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
Diagonally Implicit Runge-Kutta Methods for Ordinary Differential Equations. A Review
A review of diagonally implicit Runge-Kutta (DIRK) methods applied to rst-order ordinary di erential equations (ODEs) is undertaken. The goal of this review is to summarize the characteristics, assess the potential, and then design several nearly optimal, general purpose, DIRK-type methods. Over 20 important aspects of DIRKtype methods are reviewed. A design study is then conducted on DIRK-type methods having from two to seven implicit stages. From this, 15 schemes are selected for general purpose application. Testing of the 15 chosen methods is done on three singular perturbation problems. Based on the review of method characteristics, these methods focus on having a stage order of two, sti accuracy, L-stability, high quality embedded and dense-output methods, small magnitudes of the algebraic stability matrix eigenvalues, small values of aii, and small or vanishing values of the internal stability function for large eigenvalues of the Jacobian. Among the 15 new methods, ESDIRK4(3)6L[2]SA is recommended as a good default method for solving sti problems at moderate error tolerances
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