60,338 research outputs found
Downwinding for Preserving Strong Stability in Explicit Integrating Factor Runge--Kutta Methods
Strong stability preserving (SSP) Runge-Kutta methods are desirable when
evolving in time problems that have discontinuities or sharp gradients and
require nonlinear non-inner-product stability properties to be satisfied.
Unlike the case for L2 linear stability, implicit methods do not significantly
alleviate the time-step restriction when the SSP property is needed. For this
reason, when handling problems with a linear component that is stiff and a
nonlinear component that is not, SSP integrating factor Runge--Kutta methods
may offer an attractive alternative to traditional time-stepping methods. The
strong stability properties of integrating factor Runge--Kutta methods where
the transformed problem is evolved with an explicit SSP Runge--Kutta method
with non-decreasing abscissas was recently established. In this work, we
consider the use of downwinded spatial operators to preserve the strong
stability properties of integrating factor Runge--Kutta methods where the
Runge--Kutta method has some decreasing abscissas. We present the SSP theory
for this approach and present numerical evidence to show that such an approach
is feasible and performs as expected. However, we also show that in some cases
the integrating factor approach with explicit SSP Runge--Kutta methods with
non-decreasing abscissas performs nearly as well, if not better, than with
explicit SSP Runge--Kutta methods with downwinding. In conclusion, while the
downwinding approach can be rigorously shown to guarantee the SSP property for
a larger time-step, in practice using the integrating factor approach by
including downwinding as needed with optimal explicit SSP Runge--Kutta methods
does not necessarily provide significant benefit over using explicit SSP
Runge--Kutta methods with non-decreasing abscissas.Comment: arXiv admin note: text overlap with arXiv:1708.0259
Classification of Stochastic Runge-Kutta Methods for the Weak Approximation of Stochastic Differential Equations
In the present paper, a class of stochastic Runge-Kutta methods containing
the second order stochastic Runge-Kutta scheme due to E. Platen for the weak
approximation of It\^o stochastic differential equation systems with a
multi-dimensional Wiener process is considered. Order one and order two
conditions for the coefficients of explicit stochastic Runge-Kutta methods are
solved and the solution space of the possible coefficients is analyzed. A full
classification of the coefficients for such stochastic Runge-Kutta schemes of
order one and two with minimal stage numbers is calculated. Further, within the
considered class of stochastic Runge-Kutta schemes coefficients for optimal
schemes in the sense that additionally some higher order conditions are
fulfilled are presented
Strong Stability Preserving Integrating Factor Runge-Kutta Methods
Strong stability preserving (SSP) Runge-Kutta methods are often desired when
evolving in time problems that have two components that have very different
time scales. Where the SSP property is needed, it has been shown that implicit
and implicit-explicit methods have very restrictive time-steps and are
therefore not efficient. For this reason, SSP integrating factor methods may
offer an attractive alternative to traditional time-stepping methods for
problems with a linear component that is stiff and a nonlinear component that
is not. However, the strong stability properties of integrating factor
Runge-Kutta methods have not been established. In this work we show that it is
possible to define explicit integrating factor Runge-Kutta methods that
preserve the desired strong stability properties satisfied by each of the two
components when coupled with forward Euler time-stepping, or even given weaker
conditions. We define sufficient conditions for an explicit integrating factor
Runge--Kutta method to be SSP, namely that they are based on explicit SSP
Runge--Kutta methods with non-decreasing abscissas. We find such methods of up
to fourth order and up to ten stages, analyze their SSP coefficients, and prove
their optimality in a few cases. We test these methods to demonstrate their
convergence and to show that the SSP time-step predicted by the theory is
generally sharp, and that the non-decreasing abscissa condition is needed in
our test cases. Finally, we show that on typical total variation diminishing
linear and nonlinear test-cases our new explicit SSP integrating factor
Runge-Kutta methods out-perform the corresponding explicit SSP Runge-Kutta
methods, implicit-explicit SSP Runge--Kutta methods, and some well-known
exponential time differencing methods
Rooted Tree Analysis for Order Conditions of Stochastic Runge-Kutta Methods for the Weak Approximation of Stochastic Differential Equations
A general class of stochastic Runge-Kutta methods for the weak approximation
of It\^o and Stratonovich stochastic differential equations with a
multi-dimensional Wiener process is introduced. Colored rooted trees are used
to derive an expansion of the solution process and of the approximation process
calculated with the stochastic Runge-Kutta method. A theorem on general order
conditions for the coefficients and the random variables of the stochastic
Runge-Kutta method is proved by rooted tree analysis. This theorem can be
applied for the derivation of stochastic Runge-Kutta methods converging with an
arbitrarily high order
Positive and elementary stable explicit nonstandard Runge-Kutta methods for a class of autonomous dynamical systems
In this paper, we construct explicit nonstandard Runge-Kutta (ENRK) methods
which have higher accuracy order and preserve two important properties of
autonomous dynamical systems, namely, the positivity and linear stability.
These methods are based on the classical explicit Runge-Kutta methods, where
instead of the usual in the formulas there stands a function .
It is proved that the constructed methods preserve the accuracy order of the
original Runge-Kutta methods. The numerical simulations confirm the validity of
the obtained theoretical results
Functional Continuous Runge-Kutta Methods with Reuse
In the paper explicit functional continuous Runge-Kutta and
Runge-Kutta-Nystr\"om methods for retarded functional differential equations
are considered. New methods for first order equations as well as for second
order equations of the special form are constructed with the reuse of the last
stage of the step. The order conditions for Runge-Kutta-Nystr\"om methods are
derived. Methods of orders three, four and five which require less computations
than the known methods are presented. Numerical solution of the test problems
confirm the convergence order of the new methods and their lower computational
cost is performed.Comment: 24 pages, 16 figure
Construction of symplectic (partitioned) Runge-Kutta methods with continuous stage
Hamiltonian systems are one of the most important class of dynamical systems
with a geometric structure called symplecticity and the numerical algorithms
which can preserve such geometric structure are of interest. In this article we
study the construction of symplectic (partitioned) Runge-Kutta methods with
continuous stage, which provides a new and simple way to construct symplectic
(partitioned) Runge-Kutta methods in classical sense. This line of construction
of symplectic methods relies heavily on the expansion of orthogonal polynomials
and the simplifying assumptions for (partitioned) Runge-Kutta type methods.Comment: 13 page
An extended framework of continuous-stage Runge-Kutta methods
We propose an extended framework for continuous-stage Runge-Kutta methods
which enables us to treat more complicated cases especially for the case
weighting on infinite intervals. By doing this, various types of weighted
orthogonal polynomials (e.g., Jacobi polynomials, Laguerre polynomials, Hermite
polynomials etc.) can be used in the construction of Runge-Kutta-type methods.
Particularly, families of Runge-Kutta-type methods with geometric properties
can be constructed in this new framework. As examples, some new symplectic
integrators by using Legendre polynomials, Laguerre polynomials and Hermite
polynomials are constructed.Comment: arXiv admin note: text overlap with arXiv:1806.0338
Optimal monotonicity-preserving perturbations of a given Runge-Kutta method
Perturbed Runge--Kutta methods (also referred to as downwind Runge--Kutta
methods) can guarantee monotonicity preservation under larger step sizes
relative to their traditional Runge--Kutta counterparts. In this paper we
study, the question of how to optimally perturb a given method in order to
increase the radius of absolute monotonicity (a.m.). We prove that for methods
with zero radius of a.m., it is always possible to give a perturbation with
positive radius. We first study methods for linear problems and then methods
for nonlinear problems. In each case, we prove upper bounds on the radius of
a.m., and provide algorithms to compute optimal perturbations. We also provide
optimal perturbations for many known methods
Runge-Kutta Theory and Constraint Programming
There exist many Runge-Kutta methods (explicit or implicit), more or less
adapted to specific problems. Some of them have interesting properties, such as
stability for stiff problems or symplectic capability for problems with energy
conservation. Defining a new method suitable to a given problem has become a
challenge. The size, the complexity and the order do not stop growing. This
informal challenge to implement the best method is interesting but an important
unsolved problem persists. Indeed, the coefficients of Runge-Kutta methods are
harder and harder to compute, and the result is often expressed in
floating-point numbers, which may lead to erroneous integration schemes. Here,
we propose to use interval analysis tools to compute Runge-Kutta coefficients.
In particular, we use a solver based on guaranteed constraint programming.
Moreover, with a global optimization process and a well chosen cost function,
we propose a way to define some novel optimal Runge-Kutta methods.Comment: This is a revised version of "Runge-Kutta Theory and Constraint
Programming", Reliable Computing vol. 25, 201
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