2,525 research outputs found
On modified Runge–Kutta trees and methods
AbstractModified Runge–Kutta (mRK) methods can have interesting properties as their coefficients may depend on the step length. By a simple perturbation of very few coefficients we may produce various function-fitted methods and avoid the overload of evaluating all the coefficients in every step. It is known that, for Runge–Kutta methods, each order condition corresponds to a rooted tree. When we expand this theory to the case of mRK methods, some of the rooted trees produce additional trees, called mRK rooted trees, and so additional conditions of order. In this work we present the relative theory including a theorem for the generating function of these additional mRK trees and explain the procedure to determine the extra algebraic equations of condition generated for a major subcategory of these methods. Moreover, efficient symbolic codes are provided for the enumeration of the trees and the generation of the additional order conditions. Finally, phase-lag and phase-fitted properties are analyzed for this case and specific phase-fitted pairs of orders 8(6) and 6(5) are presented and tested
Exotic aromatic B-series for the study of long time integrators for a class of ergodic SDEs
We introduce a new algebraic framework based on a modification (called
exotic) of aromatic Butcher-series for the systematic study of the accuracy of
numerical integrators for the invariant measure of a class of ergodic
stochastic differential equations (SDEs) with additive noise. The proposed
analysis covers Runge-Kutta type schemes including the cases of partitioned
methods and postprocessed methods. We also show that the introduced exotic
aromatic B-series satisfy an isometric equivariance property.Comment: 33 page
Formal series and numerical integrators: some history and some new techniques
This paper provides a brief history of B-series and the associated Butcher
group and presents the new theory of word series and extended word series.
B-series (Hairer and Wanner 1976) are formal series of functions parameterized
by rooted trees. They greatly simplify the study of Runge-Kutta schemes and
other numerical integrators. We examine the problems that led to the
introduction of B-series and survey a number of more recent developments,
including applications outside numerical mathematics. Word series (series of
functions parameterized by words from an alphabet) provide in some cases a very
convenient alternative to B-series. Associated with word series is a group G of
coefficients with a composition rule simpler than the corresponding rule in the
Butcher group. From a more mathematical point of view, integrators, like
Runge-Kutta schemes, that are affine equivariant are represented by elements of
the Butcher group, integrators that are equivariant with respect to arbitrary
changes of variables are represented by elements of the word group G.Comment: arXiv admin note: text overlap with arXiv:1502.0552
Stochastic B-series analysis of iterated Taylor methods
For stochastic implicit Taylor methods that use an iterative scheme to
compute their numerical solution, stochastic B--series and corresponding growth
functions are constructed. From these, convergence results based on the order
of the underlying Taylor method, the choice of the iteration method, the
predictor and the number of iterations, for It\^o and Stratonovich SDEs, and
for weak as well as strong convergence are derived. As special case, also the
application of Taylor methods to ODEs is considered. The theory is supported by
numerical experiments
Effective order strong stability preserving Runge–Kutta methods
We apply the concept of effective order to strong stability preserving (SSP) explicit Runge–Kutta methods. Relative to classical Runge–Kutta methods, effective order methods are designed to satisfy a relaxed set of order conditions, but yield higher order accuracy when composed with special starting and stopping methods. The relaxed order conditions allow for greater freedom in the design of effective order methods. We show that this allows the construction of four-stage SSP methods with effective order four (such methods cannot have classical order four). However, we also prove that effective order five methods—like classical order five methods—require the use of non-positive weights and so cannot be SSP. By numerical optimization, we construct explicit SSP Runge–Kutta methods up to effective order four and establish the optimality of many of them. Numerical experiments demonstrate the validity of these methods in practice
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