A new explicit fourth-order six-stage Runge-Kutta scheme with low dispersion and low dissipation properties is developed. This new Runge-Kutta scheme is shown to be more efficient in terms of dispersion and dissipation properties than existing algorithms such as Runge-Kutta temporal schemes developed by Hu et al. (1996), Mead and Renaut (1999), Tselios and Simos (2005). We perform a spectral an...

Abstract. Space discretization of some time-dependent partial differential equations gives rise to stiff systems of ordinary differential equations. In this case, implicit methods should be used and therefore, in general, nonlinear systems must be solved. The solutions to these systems are approximated by iterative schemes and, in order to obtain an efficient code, good initializers should be u...

We construct A-stable and L-stable diagonally implicit Runge-Kutta methods of which the diagonal vector in the Butcher matrix has a minimal maximum norm. If the implicit Runge-Kutta relations are iteratively solved by means of the approximately factorized Newton process, then such iterated Runge-Kutta methods are suitable methods for integrating shallow water problems in the sense that the stab...

In this paper a new embedded Singly Diagonally Implicit Runge-Kutta Nystrom fourth order in fifth order method for solving special second order initial value problems is derived. A standard set of test problems are tested upon and comparisons on the numerical results are made when the same set of test problems are reduced to first order systems and solved using the existing embedded diagonally ...

The aim of this paper is to design a new family of numerical methods of arbitrarily high order for systems of rst-order diierential equations which are to be termed pseudo two-step Runge-Kutta methods. By using collocation techniques, we can obtain an arbitrarily high-order stable pseudo two-step Runge-Kutta method with any desired number of implicit stages in retaining the two-step nature. In ...

Implicit-explicit (IMEX) Runge-Kutta methods play a major rule in the numerical treatment of differential systems governed by stiff and non-stiff terms. This paper discusses order conditions and symplecticity properties of a class of IMEX Runge–Kutta methods in the context of optimal control problems. The analysis of the schemes is based on the continuous optimality system. Using suitable trans...

This paper investigates numerical methods for direct decoupled sensitivity and discrete adjoint sensitivity analysis of stiff systems based on implicit Runge Kutta schemes. Efficient implementations of tangent linear and adjoint schemes are discussed for two families of methods: fully implicit three-stage Runge Kutta and singly diagonally-implicit Runge Kutta. High computational efficiency is a...

Strong stability preserving (SSP) high order Runge–Kutta time discretizations were developed for use with semi-discrete method of lines approximations of hyperbolic partial differential equations, and have proven useful in many other applications. These high order time discretization methods preserve the strong stability properties of first order explicit Euler time stepping. In this paper we a...

Block Runge-Kutta formulae suitable for the approximate numerical integration of initial value problems for first order systems of ordinary differential equations are derived. Considered in detail are the problems of varying both order and stepsize automatically. This leads to a class of variable order block explicit Runge-Kutta formulae for the integration of nonstiff problems and a class of v...

Practical, structure-preserving methods for integrating classical Heisenberg spin systems are discussed. Two new integrators are derived and compared, including (1) a symmetric energy and spin-length preserving integrator based on a Red-Black splitting of the spin sites combined with a staggered timestepping scheme and (2) a (Lie-Poisson) symplectic integrator based on Hamiltonian splitting. Th...