Strong stability-preserving (SSP) Runge–Kutta methods were developed for time integration of semidiscretizations of partial differential equations. SSP methods preserve stability properties satisfied by forward Euler time integration, under a modified time-step restriction. We consider the problem of finding explicit Runge–Kutta methods with optimal SSP time-step restrictions, first for the cas...

We apply the concept of effective order to strong stability preserving (SSP) explicit Runge–Kutta methods. Relative to classical Runge–Kutta methods, methods with an effective order of accuracy are designed to satisfy a relaxed set of order conditions, but yield higher order accuracy when composed with special starting and stopping methods. We show that this allows the construction of four-stag...

A numerical method for solving Hamiltonian equations is said to be symplectic if it preserves the symplectic structure associated with the equations. Various symplectic methods are widely used in many fields of science and technology. A symplectic method preserves an approximate Hamiltonian perturbed from the original Hamiltonian. It theoretically supports the effectiveness of symplectic method...

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 ...

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...

The pioneering work of Runge and Kutta a hundred years ago has ultimately led to suites of sophisticated numerical methods suitable for solving complex systems of deterministic ordinary diierential equations. However, in many modelling situations, the appropriate representation is a stochastic diier-ential equation and here numerical methods are much less sophisticated. In this paper a very gen...

The purpose of this study is the design of efficient methods for the solution of an ordinary differential system of equations arising from the semidiscretization of a hyperbolic partial differential equation. Jameson recently introduced the use of one-step Runge-Kutta methods for the numerical solution of the Euler equations. Improvements in efficiency up to 80% may be achieved by using two-ste...

Despite the popularity of high-order explicit Runge–Kutta (ERK) methods for integrating semi-discrete systems of equations, ERK methods suffer from severe stability-based time step restrictions for very stiff problems. We implement a discontinuous Galerkin finite element method (DGFEM) along with recently introduced high-order implicit–explicit Runge–Kutta (IMEX-RK) schemes to overcome geometry...

The condition equations are derived by the introduction of a system of equivalent differential equations, avoiding the usual formalism with trees and elementary differentials. Solutions to the condition equations are found by direct numerical optimization, during which simplifying assumptions upon the Runge-Kutta coefficients may or may not be used. Depending on the optimization criterion, diff...

This paper investigates diagonally implicit Runge-Kutta methods in which the implicit relations can be solved in parallel and are singly diagonalimplicit on each processor. The algorithms are based on diagonally implicit iteration of fully implicit Runge-Kutta methods of high order. The iteration scheme is chosen in such a way that the resulting algorithm is ^(a)-stable or Z,(a)-stable with a e...