Anne Kværnø

Multirate Partitioned Runge-Kutta Methods

The coupling of subsystems in a hierarchical modelling approach leads to different time constants in the dynamical simulation of technical systems. Multirate schemes exploit the different time scales by using different time steps for the subsystems. The stiffness of the system or at least of some subsystems in chemical reaction kinetics or network analysis, for example, forbids the use of explicit integration schemes. To cope with stiff problems, we introduce multirate schemes based on partitioned Runge—Kutta methods which avoid the coupling between active and latent components based on interpolating and extrapolating state variables. Order conditions and test results for such a lower order MPRK method are presented.

Singly Diagonally Implicit Runge–Kutta Methods with an Explicit First Stage

The purpose of this paper is to construct methods for solving stiff ODEs, in particular singular perturbation problems. We consider embedded pairs of singly diagonally implicit Runge–Kutta methods with an explicit first stage (ESDIRKs). Stiffly accurate pairs of order 3/2, 4/3 and 5/4 are constructed.

B-Series Analysis of Stochastic Runge-Kutta Methods That Use an Iterative Scheme to Compute Their Internal Stage Values

In recent years, implicit stochastic Runge-Kutta (SRK) methods have been developed both for strong and weak approximations. For these methods, the stage values are only given implicitly. However, in practice these implicit equations are solved by iterative schemes such as simple iteration, modified Newton iteration or full Newton iteration. We employ a unifying approach for the construction of stochastic B-series which is valid both for It o- and Stratonovich-stochastic differential equations (SDEs) and applicable both for weak and strong convergence to analyze the order of the iterated Runge-Kutta method. Moreover, the analytical techniques applied in this paper can be of use in many other similar contexts.

The use of Butcher series in the analysis of Newton-like iterations in Runge-Kutta formulas

Abstract We consider the numerical solution of initial value problems for both ordinary differential equations and differential-algebraic equations by Runge-Kutta (RK) formulas. We assume that the internal stage values of the RK formula are computed by some iterative scheme for solving nonlinear equations, such as Newtons method. Using Butcher series and rooted trees, we give a complete characterization of the local error in the RK formula after k iterations of the scheme. Results are given for three specific schemes: simple iteration, the modified Newton iteration, and the full Newton iteration. The ideas developed in this paper, however, are easily applied to other iterative schemes of this kind.

A Time-Reversible, Regularized, Switching Integrator for the N -Body Problem

This article describes a gravitational N-body integration algorithm conserving linear and angular momentum and time-reversal symmetry. Forces are dynamically partitioned based on interbody separation, so that the long-range forces are evaluated relatively rarely, and close approaches are treated by an efficient regularization technique. The method incorporates an automatic stepsize adjustment based on a Sundman time-transformation. Although the scheme is formally second order, the most intensive computations (the close-approach dynamics) are executed at higher order, thus improving the overall accuracy. Numerical experiments indicate that the method can effectively treat few-body gravitational problems with two-body close approaches, and it compares favorably with other schemes presented in the literature.

An analysis of the order of Runge-Kutta methods that use an iterative scheme to compute their internal stage values

We employ B-series to analyze the order of Runge-Kutta (RK) methods that use simple iteration, modified Newton iteration or full Newton iteration to compute their internal stage values. Our assumptions about the initial guess for the internal stage values cover a wide range of practical schemes. Moreover, the analytical techniques developed in this paper can be applied in many other similar contexts.

Runge-Kutta research in Trondheim

Abstract Runge-Kutta research in Trondheim began in 1970 when Syvert P. Norsett was appointed to the NTH. Although the group has worked on various aspects of Runge-Kutta methods, we have elected, in this paper, to focus on DIRK methods, linear stability, order stars, parallel methods and continuous explicit methods.

Order conditions for stochastic Runge–Kutta methods preserving quadratic invariants of Stratonovich SDEs

Abstract In this paper we prove that for a stochastic Runge–Kutta method, the conditions for preserving quadratic invariants work as simplifying assumptions. For such methods, the method coefficients only have to satisfy one condition for each unrooted tree. This is a generalization of the result obtained for deterministic Runge–Kutta methods by Sanz-Serna and Abia in 1991.

Composition of stochastic B-series with applications to implicit Taylor methods

In this article, we construct a representation formula for stochastic B-series evaluated in a B-series. This formula is used to give for the first time the order conditions of implicit Taylor methods in terms of rooted trees. Finally, as an example we apply these order conditions to derive in a simple manner a family of strong order 1.5 Taylor methods applicable to Ito SDEs.

Stochastic Taylor Expansions: Weight Functions of B-Series Expressed as Multiple Integrals

The exact solution of stochastic differential equations can be expressed as stochastic B-series. In this article, we present an algorithm using rooted trees for expanding the weight functions occurring in this representation in terms of multiple integrals using multi-indices. This results in an alternative approach to express relations between multiple integrals.