Introduction PACT Abstract A parallel implementation for multi-implicit Runge-Kutta methods with real eigen-values is described. The parallel method is analysed and the algorithm is devised. For the problem with d domains, the amount within the s-stage Runge-Kutta method, associated with the solution of system, is proportional to (sd) 3. The proposed parallelisation transforms the above system ...

In this paper we are concerned with the development of an explicit Runge-Kutta scheme for the numerical solution of delay diierential equations (DDEs) where one or more delay lies in the current Runge-Kutta interval. The scheme presented is also applicable to the numerical solution of Volterra functional equations (VFEs), although the theory is not covered in this paper. We also derive the stab...

In Burrage and Burrage (1996) it was shown that by introducing a very general formulation for stochastic Runge-Kutta methods, the previous strong order barrier of order one could be broken without having to use higher derivative terms. In particular, methods of strong order 1.5 were developed in which a Stratonovich integral of order one and one of order two were present in the formulation. In ...

We describe the derivation of order conditions, without restrictions on stage order, for general linear methods for ordinary differential equations. This derivation is based on the extension of Albrecht approach proposed in the context of Runge-Kutta and composite and linear cyclic methods. This approach was generalized by Jackiewicz and Tracogna to two-step Runge-Kutta methods, by Jackiewicz a...

The implementation of implicit Runge{Kutta methods requires the solution of large systems of non-linear equations. Normally these equations are solved by a modiied Newton process, which can be very expensive for problems of high dimension. The recently proposed triangularly implicit iteration methods for ODE-IVP solvers 5] substitute the Runge{Kutta matrix A in the Newton process for a triangul...

In this paper, we analyze the stability of the fourth order Runge-Kutta method for integrating semi-discrete approximations of time-dependent partial differential equations. Our study focuses on linear problems and covers general semi-bounded spatial discretizations. A counter example is given to show that the classical four-stage fourth order Runge-Kutta method can not preserve the one-step st...

New Runge–Kutta methods for method of lines solution of systems of ordinary differential equations arising from discretizations of spatial derivatives in hyperbolic equations, by Chebyshev or modified Chebyshev methods, are introduced. These Runge–Kutta methods optimize the time step necessary for stable solutions, while holding dispersion and dissipation fixed. It is found that maximizing disp...

Abstract. We investigate the strong convergence rate of both Runge–Kutta methods and simplified step-N Euler schemes for stochastic differential equations driven by multi-dimensional fractional Brownian motions with H ∈ ( 2 , 1). These two classes of numerical schemes are implementable in the sense that the required information from the driving noises are only their increments. We prove the sol...

The optimization of some W-methods [7] for the time integration of time-dependent PDEs in several spatial variables is considered. In [2, Theorem 1] several three-parametric families of three-stage W-methods for the integration of IVPs in ODEs were studied. Besides, the optimization of several specific methods for PDEs when the Approximate Matrix Factorization Splitting (AMF) [3, 4] is used to ...

Standard Runge-Kutta methods are explicit, one-step, and generally constant step-size numerical integrators for the solution of initial value problems. Such integration schemes of orders 3, 4, and 5 require 3, 4, and 6 function evaluations per time step of integration, respectively. In this paper, we propose a set of simple, explicit, and constant step-size Accerelated-Runge-Kutta methods that ...