The main purpose of this paper is to discuss the equilibrium attractive properties of a class of multistep Runge–Kutta methods for initial value problems of ordinary differential equations. Some algebraic conditions insuring the equilibrium attractivity are given, and some methods satisfying these algebraic conditions are constructed. Some numerical examples confirm our results. 2005 Elsevier I...

A personal view of the historical development of software based on explicit Runge-Kutta formulas is presented, and some future developments are suggested.

This work constructs a new class of multirate schemes based on the recently developed generalized additive Runge-Kutta (GARK) methods [10]. Multirate schemes use different step sizes for different components and for different partitions of the right-hand side based on the local activity levels. We show that the new multirate GARK family includes many well-known multirate schemes as special case...

For Hamiltonian systems with non-canonical structure matrix a new class of numerical integrators is proposed. The methods exactly preserve energy, are invariant with respect to linear transformations, and have arbitrarily high order. Those of optimal order also preserve quadratic Casimir functions. The discussion of the order is based on an interpretation as partitioned Runge–Kutta method with ...

This paper deals with the stability of Runge–Kutta methods for a class of stiff systems of nonlinear Volterra delay-integro-differential equations. Two classes of methods are considered: Runge–Kutta methods extended with a compound quadrature rule, and Runge– Kutta methods extended with a Pouzet type quadrature technique. Global and asymptotic stability criteria for both types of methods are de...

Article history: Received 27 April 2010 Received in revised form 2 August 2010 Accepted 19 August 2010 Available online 26 August 2010

In this work, we derive the order conditions for fourth order time splitting schemes in the case of the 1D Vlasov-Poisson system. Computations to obtain such conditions are motivated by the specific Poisson structure of the Vlasov-Poisson system : this structure is similar to Runge-Kutta-Nyström systems. The obtained conditions are proved to be the same as RKN conditions derived for ODE up to t...

Spectral deferred correction is a flexible technique for constructing high-order, stiffly-stable time integrators using a low order method as a base scheme. Here we examine their use in conjunction with splitting methods to solve initial-boundary value problems for partial differential equations. We exploit their close connection with implicit Runge–Kutta methods to prove that up to the full ac...

Maŕıa López-Fernández1, Christian Lubich2, Cesar Palencia1, and Achim Schädle3 1 Departamento de Matemática Aplicada y Computación, Universidad de Valladolid, Valladolid, Spain. E-mail: {marial, palencia}@mac.cie.uva.es 2 Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D–72076 Tübingen, Germany. E-mail: [email protected] 3 ZIB Berlin, Takustr. 7, D-14195 Berlin,...